notes polynomials and nonlinear introduction functions · • simplify expressions containing...

IntroductionIn this unit, students will be intro-duced to nonlinear functions.Students will first learn aboutpolynomials and operationsinvolving monomials and poly-nomials. Students then learn var-ious methods of factoring, andare finally introduced to quad-ratic and exponential functions.

Assessment OptionsUnit 3 Test Pages 641–642

of the Chapter 10 Resource Mastersmay be used as a test or reviewfor Unit 3. This assessment con-tains both multiple-choice andshort answer items.

TestCheck andWorksheet Builder

This CD-ROM can be used tocreate additional unit tests andreview worksheets.

Chapter 8Polynomials

Chapter 9Factoring

Chapter 10Quadratic and Exponential Functions

Polynomialsand NonlinearFunctions

Polynomialsand NonlinearFunctionsNot all real-world

situations can bemodeled using alinear function. In thisunit, you will learnabout polynomialsand nonlinearfunctions.

406 Unit 3 Polynomials and Nonlinear Functions

406 Unit 3 Polynomials and Nonlinear Functions

NotesNotes

Source: USA TODAY, March 28, 2001

“Like any former third-grader, Catherine Beyhl

knows that the solar system has nine planets, and she

knows a phrase to help remember their order: ‘My

Very Educated Mother Just Served Us Nine Pizzas.’

But she recently visited the American Museum of

Natural History’s glittering new astronomy hall at the

Hayden Planetarium and found only eight scale

models of the planets. No Pizza—no Pluto.” In this

project, you will examine how scientific notation,

factors, and graphs are useful in presenting

information about the planets.

Then continue working

on your WebQuest as

you study Unit 3.

Log on to www.algebra1.com/webquest.Begin your WebQuest by reading the Task.

Pluto Is Falling From Status as Distant Planet

Unit 3 Polynomials and Nonlinear Functions 407

8-3 9-1 10-2

429 479 537

LessonPage

Are we alone in the universe?Adults who believe that during the next centuryevidence will be discovered that shows:

By Cindy Hall and Sam Ward, USA TODAY

Source: The Gallup Organization for the John Templeton Foundation

66%Don’tknow

28%

6%Other lifein this orothergalaxies

Life existsonly onEarth

USA TODAY Snapshots®

Internet Project A WebQuest is an online project in which students do research on the Internet,gather data, and make presentations using word processing, graphing, page-making, or presentation software. In each chapter, students advance tothe next step in their WebQuest. At the end of Chapter 10, the project culminates with a presentation of their findings.

Teaching notes and sample answers are available in the WebQuest and Project Resources.

TeachingSuggestions

Have students study the USA TODAY Snapshot.• Ask students what fraction

of people believe that evi-dence will be discovered thatlife exists in this or othergalaxies.

• Take an informal poll in theclass and compare the resultsto the USA TODAY results.

• Point out to students that intheir WebQuest they will bedesigning a display about theplanets in the solar system.

Additional USA TODAYSnapshots appearing in Unit 3:

Chapter 8 A sweet holidayseason (p. 427)

Chapter 9 Number of domainregistrationsclimbs (p. 494)

Chapter 10 Spending more oneating out (p. 561)

Making gains (p. 563)

Grand CanyonVisitors (p. 564)

2�3

Unit 3 Polynomials and Nonlinear Functions 407

http://www.algebra1.com/webquest

408A Chapter 8 Polynomials

Pacing suggestions for the entire year can be found on pages T20–T21.

PolynomialsChapter Overview and PacingPolynomialsChapter Overview and Pacing

PACING (days)Regular Block

Basic/ Basic/ Average Advanced Average Advanced

Multiplying Monomials (pp. 410–416) 2 2 0.5 0.5• Multiply monomials. (with 8-1 (with 8-1• Simplify expressions involving powers of monomials. Follow-Up) Follow-Up)Follow-Up: Use paper prisms to investigate surface area and volume.

Dividing Monomials (pp. 417–423) 2 2 1.5 1.5• Simplify expressions involving the quotient of monomials. (with 8-1 (with 8-1• Simplify expressions containing negative exponents. Follow-Up) Follow-Up)

Scientific Notation (pp. 425–430) 1 1 0.5 0.5• Express numbers in scientific notation and standard notation.• Find products and quotients of numbers expressed in scientific notation.

Polynomials (pp. 431–436) 2 1 1 0.5Preview: Use algebra tiles to model polynomials. (with 8-4 (with 8-4• Find the degree of a polynomial. Preview) Preview)• Arrange the terms of a polynomial in ascending or descending order.

Adding and Subtracting Polynomials (pp. 437–443) 2 2 1 1Preview: Use algebra tiles to add and subtract polynomials.• Add polynomials.• Subtract polynomials.

Multiplying a Polynomial by a Monomial (pp. 444–449) 2 1 1 0.5• Find the product of a monomial and a polynomial.• Solve equations involving polynomials.

Multiplying Polynomials (pp. 450–457) 2 2 1 1Preview: Use algebra tiles to find the product of two binomials. (with 8-7 (with 8-7• Multiply two binomials by using the FOIL method. Preview) Preview)• Multiply two polynomials by using the Distributive Property.

Special Products (pp. 458–463) 1 1 0.5 0.5• Find squares of sums and differences.• Find the product of a sum and a difference.

Study Guide and Practice Test (pp. 464–469) 1 1 0.5 0.5Standardized Test Practice (pp. 470–471)

Chapter Assessment 1 1 0.5 0.5

TOTAL 16 14 8 7

LESSON OBJECTIVES

*Key to Abbreviations: GCS � Graphing Calculator and Speadsheet Masters,SC � School-to-Career Masters, SM � Science and Mathematics Lab Manual

Study Guide and Intervention, Skills Practice, Practice, and Parent and Student Study Guide Workbooks are also available in Spanish.

ELL

Chapter 8 Polynomials 408B

Materials

455–456 457–458 459 460 59 8-1 8-1 18 (Follow-Up: centimetergrid paper)

461–462 463–464 465 466 517 60 8-2 8-2 19 graphing calculator

467–468 469–470 471 472 33–36 SC 15, 61 8-3 8-3 20 graphing calculator SM 67–70

473–474 475–476 477 478 517, 519 GCS 37 62 8-4 8-4 (Preview: algebra tiles)

479–480 481–482 483 484 63 8-5 8-5 (Preview: algebra tiles)

485–486 487–488 489 490 518 SC 16 64 8-6 8-6 21

491–492 493–494 495 496 GCS 38 65 8-7 8-7 22 (Preview: algebra tiles,product mat)

497–498 499–500 501 502 518 66 8-8 8-8 23

503–516, 67520–522

Chapter Resource ManagerChapter Resource ManagerSt

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CHAPTER 8 RESOURCE MASTERS

Inte

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Chal

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Timesaving Tools

All-In-One Planner and Resource Center

See pages T12–T13.

™

408C Chapter 8 Polynomials

Mathematical Connections and BackgroundMathematical Connections and Background

Multiplying MonomialsA monomial is a number, variable, or a product of

a number and one or more variables. Monomials do notinvolve the addition, subtraction, or division of variables.Monomials that only involve real numbers are called con-stants. This is because their value does not change.

When multiplying monomials, use theCommutative and Associative Properties to group con-stants together and group powers with the same basetogether. To multiply powers with the same base, addthe exponents. Multiply the exponents to find a power ofa power. When finding the power of a product, find thepower of each factor. A monomial expression is simpli-fied when each base appears exactly once, there are nopowers of powers, and all fractions are in simplest form.

Dividing MonomialsExponents are subtracted when dividing two pow-

ers that have the same base. To find the power of a quo-tient, find the power of both the numerator and thedenominator. If the numerator and denominator of a frac-tion are the same, the value of the fraction is 1. Therefore,if both the numerator and denominator have the samebase raised to the same exponent, the value of the fractionis 1. Using the Quotient of Powers Property, the exponentsof the original fraction are subtracted and the fractionsimplifies to the base raised to the zero power. So it fol-lows that any base raised to the zero power equals one.

A positive number raised to a negative powerrepresents the reciprocal of the number with the oppo-site or positive exponent. Likewise, if a negative expo-nent appears in the denominator of a fraction, thispower is equivalent to a fraction with this same baseraised to the opposite power in its numerator.

Scientific NotationWhen a number is written in scientific notation, the

power of 10 is the number of places the decimal moves. Toconvert a number that is in scientific notation to one instandard form, move the decimal the number of placesindicated by the exponent. A positive exponent is used torepresent a number that is greater than or equal to 10 instandard form. A negative exponent represents a numberthat is less than 1 in standard form. To translate a numberin standard form to scientific notation, first move thedecimal to the right of the first non-zero digit. Then writethe appropriate power of 10 to the right of the number.Keep in mind what the sign of the power indicates.

Numbers in scientific notation can be multiplied ordivided. First multiply or divide the decimals. Then apply

Prior KnowledgePrior KnowledgeIn Chapter 1, students apply the order ofoperations, and the Distributive, Commuta-tive, and Associative Properties to simplifyexpressions. Students perform operations

with real numbers in Chapter 2.

This Chapter

Future ConnectionsFuture ConnectionsStudents will apply their knowledge of multi-plying and dividing monomials and polyno-mials to factoring polynomials in Chapter 9.The concepts are essential to simplifying andsolving many problems involving upper levelmathematics and science.

Continuity of InstructionContinuity of Instruction

This ChapterThis chapter helps students master opera-tions with monomials and polynomials. Itbegins with connecting the multiplicationand division of monomials with variablesand exponents to multiplying and dividingreal numbers. Students convert numbersfrom standard notation to scientific nota-tion, and vice versa, then multiply and dividenumbers written in scientific notation.Polynomials are defined, and students learnto perform operations with them. Finally,students learn patterns for finding products

of some special polynomials.

Chapter 8 Polynomials 408D

either the Product of Powers or the Quotient of PowersProperty to simplify the powers of 10. Next, rewrite thedecimal in scientific notation and simplify the powersof 10. The result is in scientific notation, but it can alsobe presented in standard form if preferred.

PolynomialsA polynomial is a monomial, or a sum of

monomials. Remember that subtraction can be rewrit-ten as addition. A monomial has only one term and isconsidered a type of polynomial. Some polynomialsthat contain more than one monomial, or term, havespecial names. A binomial has two terms and a trino-mial has three terms. All others are just referred to aspolynomials. The degree of a monomial is the sum ofthe exponents of all its variables. The term with thegreatest degree determines the degree of a polynomi-al. Usually the terms of a polynomial are arranged sothat the powers of one variable are in ascending(increasing) or descending (decreasing) order. Thisaids in reading and understanding the polynomial.

Adding and SubtractingPolynomialsTo add polynomials, combine like terms. Like

terms have the same variable bases with the sameexponents. The coefficients of like terms are addedusing the rules for adding real numbers. The basesand exponents of these terms, however, remain thesame. The rule for subtracting polynomials is thesame as the rule for subtracting integers. First replaceeach term of the second polynomial with its additiveinverse or opposite. Then combine like terms usingthe rules for adding real numbers.

Multiplying a Polynomial by a MonomialTo multiply a polynomial by a monomial,

apply the Distributive Property by multiplying eachterm in the polynomial by the monomial. Use therules for multiplying monomials. If the monomial isnegative, don’t forget to apply the rules for multiply-ing real numbers. Be sure to simplify by combiningany like terms.

Equations may contain polynomials. To solvethese equations, first simplify each side using theorder of operations by multiplying, adding, and sub-tracting as indicated. Then apply the rules for solvingmulti-step equations and equations with variables onboth sides.

Multiplying PolynomialsThe Distributive Property is applied twice

when multiplying two binomials. Multiply the firstterm of the first binomial by each term of the secondbinomial. Do the same with the second term of thefirst binomial. Then combine like terms. This resultsin a multiplying pattern called the FOIL method. Youmultiply the First terms, Outer terms, Inside terms,and Last terms of the binomials. The DistributiveProperty is used to multiply any two polynomials.The product is not in simplest terms until all liketerms have been combined.

Special ProductsThe Distributive Property and FOIL method can

always be used to multiply polynomials. However,some binomial products have patterns that make theirmultiplication simpler. One product involves the mul-tiplying of two identical binomials, called the square ofa sum. The pattern is (a � b)(a � b) � a2 � 2ab � b2.There is also a pattern for the square of a difference:(a � b)(a � b) � a2 � 2ab � b2. Note that the only dif-ference in the two patterns is the sign before the mid-dle term. A third pattern exists for the product of asum and a difference: (a � b)(a � b) � a2 � b2. Theproducts of the Outer terms and the Inner terms add tozero. While it is not essential to learn these patterns,identifying when to use them can make simplifyingthese products quicker and less laborious.

Additional mathematical information and teaching notesare available in Glencoe’s Algebra 1 Key Concepts:Mathematical Background and Teaching Notes, which is available at www.algebra1.com/key_concepts. The lessons appropriate for this chapter are as follows.• Multiplying Monomials (Lesson 22)• Dividing Monomials (Lesson 23)• Adding and Subtracting Polynomials (Lesson 24)• Multiplying a Polynomial by a Monomial (Lesson 25)• Multiplying Polynomials (Lesson 26)

www.algebra1.com/key_concepts

http://www.algebra1.com/key_concepts

408E Chapter 8 Polynomials

TestCheck and Worksheet BuilderThis networkable software has three modules for interventionand assessment flexibility:• Worksheet Builder to make worksheet and tests• Student Module to take tests on screen (optional)• Management System to keep student records (optional)

Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests.

Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters

Ongoing Prerequisite Skills, pp. 409, 415,423, 430, 436, 443, 449, 457

Practice Quiz 1, p. 430Practice Quiz 2, p. 449

AlgePASS: Tutorial Pluswww.algebra1.com/self_check_quizwww.algebra1.com/extra_examples

5-Minute Check TransparenciesPrerequisite Skills Workbook, pp. 33–36Quizzes, CRM pp. 517–518Mid-Chapter Test, CRM p. 519Study Guide and Intervention, CRM pp. 455–456,

461–462, 467–468, 473–474, 479–480, 485–486,491–492, 497–498

MixedReview

Cumulative Review, CRM p. 520 pp. 415, 423, 430, 436, 443,449, 457, 463

ErrorAnalysis

Find the Error, TWE pp. 413, 421Unlocking Misconceptions, TWE pp. 421, 433Tips for New Teachers, TWE pp. 426, 459

Find the Error, pp. 413, 421, 441Common Misconceptions,

pp. 420, 432, 454

StandardizedTest Practice

TWE pp. 470–471Standardized Test Practice, CRM pp. 521–522

Standardized Test Practice CD-ROM

www.algebra1.com/standardized_test

pp. 415, 420, 421, 423, 430,436, 443, 448, 457, 463, 469,470–471

Open-EndedAssessment

Modeling: TWE pp. 415, 436, 449Speaking: TWE pp. 423, 457Writing: TWE pp. 430, 443, 463Open-Ended Assessment, CRM p. 515

Writing in Math, pp. 415, 423,430, 436, 443, 448, 457, 463

Open Ended, pp. 413, 421, 428,434, 441, 446, 455, 461

Standardized Test, p. 471

ChapterAssessment

Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 503–508

Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 509–514

Vocabulary Test/Review, CRM p. 516

TestCheck and Worksheet Builder(see below)

MindJogger Videoquizzes www.algebra1.com/

vocabulary_reviewwww.algebra1.com/chapter_test

Study Guide, pp. 464–468Practice Test, p. 469

Additional Intervention ResourcesThe Princeton Review’s Cracking the SAT & PSATThe Princeton Review’s Cracking the ACTALEKS

and Assessmentand AssessmentA

SSES

SMEN

TIN

TER

VEN

TIO

N

Type Student Edition Teacher Resources Technology/Internet

http://www.algebra1.com/self_check_quiz

http://www.algebra1.com/extra_examples

http://www.algebra1.com/standardized_test

http://www.algebra1.com/vocabulary_review

http://www.algebra1.com/chapter_test

Chapter 8 Polynomials 408F

Algebra 1Lesson

AlgePASS Lesson

8-1 18 Performing Operations with Exponents II

8-2 19 Performing Operations with Exponents I

8-3 20 Using Scientific Notation

8-6 21 Simplifying Polynomial Expressions

8-7 22 Multiplying Binomial Expressions I

8-8 23 Multiplying Binomial Expressions II

ALEKS is an online mathematics learning system thatadapts assessment and tutoring to the student’s needs.Subscribe at www.k12aleks.com.

For more information on Reading and Writing inMathematics, see pp. T6–T7.

Intervention at HomeParent and Student Study Guide Parents and students may work together to reinforce theconcepts and skills of this chapter. (Workbook, pp. 59–67 or log on to www.algebra1.com/parent_student)

Intervention TechnologyAlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum.

Reading and Writingin Mathematics

Reading and Writingin Mathematics

Glencoe Algebra 1 provides numerous opportunities toincorporate reading and writing into the mathematics classroom.

Student Edition

• Foldables Study Organizer, p. 409• Concept Check questions require students to verbalize

and write about what they have learned in the lesson.(pp. 413, 421, 428, 434, 441, 446, 455, 461)

• Reading Mathematics, p. 424 • Writing in Math questions in every lesson, pp. 415, 423,

430, 436, 443, 448, 457, 463• Reading Study Tip, pp. 410, 425• WebQuest, p. 429

Teacher Wraparound Edition

• Foldables Study Organizer, pp. 409, 464• Study Notebook suggestions, pp. 413, 416, 421, 424,

428, 431, 434, 438, 446, 451, 455, 461 • Modeling activities, pp. 415, 436, 449• Speaking activities, pp. 423, 457• Writing activities, pp. 430, 443, 463• Differentiated Instruction, (Verbal/Linguistic), p. 460• Resources, pp. 408, 414, 422, 424, 429, 435,

442, 448, 456, 460, 462, 464

Additional Resources

• Vocabulary Builder worksheets require students todefine and give examples for key vocabulary terms asthey progress through the chapter. (Chapter 8 ResourceMasters, pp. vii-viii)

• Reading to Learn Mathematics master for each lesson(Chapter 8 Resource Masters, pp. 459, 465, 471, 477,483, 489, 495, 501)

• Vocabulary PuzzleMaker software creates crossword,jumble, and word search puzzles using vocabulary liststhat you can customize.

• Teaching Mathematics with Foldables provides suggestions for promoting cognition and language.

• Reading and Writing in the Mathematics Classroom• WebQuest and Project Resources• Hot Words/Hot Topics Sections 2.1, 3.1–3.4, 6.2, 6.3,

7.5, 7.7, 7.8

ELL

For more information on Intervention andAssessment, see pp. T8–T11.

Log on for student study help.• For each lesson in the Student Edition, there are Extra

Examples and Self-Check Quizzes.www.algebra1.com/extra_exampleswww.algebra1.com/self_check_quiz

• For chapter review, there is vocabulary review, test practice, and standardized test practice.www.algebra1.com/vocabulary_reviewwww.algebra1.com/chapter_testwww.algebra1.com/standardized_test

http://www.k12aleks.com

http://www.algebra1.com/parent_student






Have students read over the listof objectives and make a list ofany words with which they arenot familiar.

Point out to students that this isonly one of many reasons whyeach objective is important.Others are provided in theintroduction to each lesson.

Polynomials

• monomial (p. 410)• scientific notation (p. 425)• polynomial (p. 432)• binomial (p. 432)• FOIL method (p. 453)

Key Vocabulary• Lessons 8-1 and 8-2 Find products and

quotients of monomials.

• Lesson 8-3 Express numbers in scientific andstandard notation.

• Lesson 8-4 Find the degree of a polynomialand arrange the terms in order.

• Lessons 8-5 through 8-7 Add, subtract, andmultiply polynomial expressions.

• Lesson 8-8 Find special products of binomials.

Operations with polynomials, including addition, subtraction, and multiplication, form the foundation for solving equations that involve polynomials. In addition, polynomials are used to model many real-world situations. In Lesson 8-6, you

will learn how to find the distance that runners

on a curved track should be staggered.

408 Chapter 8 Polynomials408 Chapter 8 Polynomials

408 Chapter 8 Polynomials

NotesNotes

NCTM LocalLesson Standards Objectives

8-1 2, 6, 8, 9, 10

8-1 2, 3, 6, 7Follow-Up

8-2 2, 6, 8, 9, 10

8-3 1, 6, 8, 9, 10

8-4Preview 2, 10

8-4 2, 6, 8, 9, 10

8-5 2, 10Preview

8-5 2, 6, 8, 9, 10

8-6 2, 6, 8, 9, 10

8-7 2, 10Preview

8-7 2, 6, 8, 9, 10

8-8 2, 6, 8, 9, 10

Key to NCTM Standards: 1=Number & Operations, 2=Algebra,3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=ProblemSolving, 7=Reasoning & Proof,8=Communication, 9=Connections,10=Representation

Vocabulary BuilderThe Key Vocabulary list introduces students to some of the main vocabulary termsincluded in this chapter. For a more thorough vocabulary list with pronunciations ofnew words, give students the Vocabulary Builder worksheets found on pages vii andviii of the Chapter 8 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they addthese sheets to their study notebooks for future reference when studying for theChapter 8 test.

ELL

This section provides a review ofthe basic concepts needed beforebeginning Chapter 8. Page refer-ences are included for additionalstudent help.

Additional review is provided inthe Prerequisite Skills Workbook,pp. 33–36.

Prerequisite Skills in the GettingReady for the Next Lesson sectionat the end of each exercise setreview a skill needed in the nextlesson.

Make this Foldable to help you organize information aboutpolynomials. Begin with a sheet of 11" by 17" paper.

Mon.

Poly.

Draw lines along folds and label

as shown.

Fold a 2" tab along the width.

Then fold the rest in fourths.

Fold in thirdslengthwise.

Reading and Writing As you read and study the chapter, write examples and notesfor each operation.

Prerequisite Skills To be successful in this chapter, you’ll need to masterthese skills and be able to apply them in problem-solving situations. Reviewthese skills before beginning Chapter 8.

For Lessons 8-1 and 8-2 Exponential Notation

Write each expression using exponents. (For review, see Lesson 1-1.)

1. 2 � 2 � 2 � 2 � 2 25 2. 3 � 3 � 3 � 3 34 3. 5 � 5 52 4. x � x � x x3

5. a � a � a � a � a � a a6 6. x � x � y � y � y x2y3 7. �12

� � �12

� � �12

� � �12

� � �12

� ��12

��58. �

ba

� � �ba

� � �dc

� � �dc

� � �dc

�

For Lessons 8-1 and 8-2 Evaluating Powers

Evaluate each expression. (For review, see Lesson 1-1.)

9. 32 9 10. 43 64 11. 52 25 12. 104 10,00013. (�6)2 36 14. (�3)3 �27 15. ��

23

��4�18

61� 16. ��

78

��2�46

94�

For Lessons 8-1, 8-2, and 8-5 through 8-8 Area and Volume

Find the area or volume of each figure shown below. (For review, see pages 813–817.)

17. 18. 19. 20.

5 cm 5 cm

5 cm

4 ft

3 ft7 ft

6m9 yd

Fold

Label

Open and Fold

��ab

��2��dc

��3

63 yd2

36� m2

or about113.04 m2

84 ft3

125 cm3

Chapter 8 Polynomials 409Chapter 8 Polynomials 409

Chapter 8 Polynomials 409

For PrerequisiteLesson Skill

8-2 Simplifying Fractions (p. 415)

8-3 Products of Powers (p. 423)

8-4 Evaluating Expressions (p. 430)

8-5 Simplifying Expressions (p. 436)

8-6 Distributive Property (p. 443)

8-7 Products of Powers (p. 449)

8-8 Power of a Power, Power of aProduct (p. 457)

Organization of Data using a Table to Make ComparisonsAfter students make their Foldable tables, have them label thecolumns and rows as illustrated. Students use their Foldables to takenotes and write examples for each operation. Have students comparedifferent functions. For example, compare adding monomials andadding polynomials. Remind students that comparing involvesdetermining a trait to be compared and then finding the similaritiesand differences in that trait.

TM

For more informationabout Foldables, seeTeaching Mathematicswith Foldables.

5-Minute CheckTransparency 8-1 Use as a

quiz or review of Chapter 7.

Mathematical Background notesare available for this lesson on p. 408C.

does doubling speedquadruple braking

distance?Ask students:• What does the term quadrupled

mean? increased by 4 times, ormultiplied by 4

• Why isn’t the braking distance 4times the speed? The expressionfor finding the braking distance,

where s is speed in mph, is s2,not 4s.

• Based on the chart, what wouldbe the braking distance for acar traveling 80 mph? 320 ft120 mph? 720 ft

• Drag Racing Suppose a sportscar on a drag strip can reach100 miles per hour in a quarterof a mile. Using the brakingdistance table, calculate howfar the car would travel on thedrag strip, from start to stop, ifthe driver started braking whenthe car reached 100 miles perhour. A mile is 5280 feet. 1820 ft

1�20


MULTIPLY MONOMIALS An expression like �210�s2 is called a monomial.

A is a number, a variable, or a product of a number and one or morevariables. An expression involving the division of variables is not a monomial.Monomials that are real numbers are called .constants

monomial

Recall that an expression of the form xn is called a power and represents theproduct you obtain when x is used as a factor n times. The number x is the base,and the number n is the exponent.

25 � 2 � 2 � 2 � 2 � 2 or 32

In the following examples, the definition of a power is used to find the productsof powers. Look for a pattern in the exponents.

Vocabulary• monomial• constant

Multiplying Monomials

Identify MonomialsDetermine whether each expression is a monomial. Explain your reasoning.

a.

b.

c.

d.

e.

Example 1Example 1

Reading MathThe expression xn is readx to the nth power.

Study Tip

Expression Monomial? Reason

�5 yes �5 is a real number and an example of a constant.

p � q noThe expression involves the addition, not theproduct, of two variables.

x yes Single variables are monomials.

�dc� no

The expression is the quotient, not the product, oftwo variables.

�ab5c8� yes

�ab

5c8� � �

15

�abc8. The expression is the product of a

number, �15

�, and three variables.

5 factorsexponent

base

←

←

�does doubling speed quadruple braking distance?does doubling speed quadruple braking distance?

The table shows the braking distance for a vehicle at certain speeds. If s represents the speed in miles per hour, then the approximate number of feet that the driver must

apply the brakes is �210�s2. Notice that

when speed is doubled, the brakingdistance is quadrupled.

• Multiply monomials.

• Simplify expressions involving powers of monomials.

Source: British Highway Code

Spee

d (m

iles

per h

our)

Braking Distance (feet)

20 20

30

40

50

60

70

45

80

125

180

245

LessonNotes

1 Focus1 Focus

Chapter 8 Resource Masters• Study Guide and Intervention, pp. 455–456• Skills Practice, p. 457• Practice, p. 458• Reading to Learn Mathematics, p. 459• Enrichment, p. 460

Parent and Student Study GuideWorkbook, p. 59

5-Minute Check Transparency 8-1Answer Key Transparencies

TechnologyAlgePASS: Tutorial Plus, Lesson 18Interactive Chalkboard

Workbook and Reproducible Masters

Resource ManagerResource Manager

Transparencies

Lesson 8-1 Multiplying Monomials 411www.algebra1.com/extra_examples

Product of Powers

Power of a Power

• Words To multiply two powers that have the same base, add the exponents.

• Symbols For any number a and all integers m and n, am � an � am � n.

• Example a4 � a12 � a4 � 12 or a16

• Words To find the power of a power, multiply the exponents.

• Symbols For any number a and all integers m and n, (am)n � am � n.

• Example (k5)9 � k5 � 9 or k45

Power of 1Recall that a variable withno exponent indicated canbe written as a power of1. For example, x � x1

and ab � a1b1.

Study Tip

Look Back To review using acalculator to find apower of a number, see Lesson 1-1.

Study Tip

Example 2Example 2 Product of PowersSimplify each expression.

a. (5x7)(x6)

(5x7)(x6) � (5)(1)(x7 � x6) Commutative and Associative Properties

� (5 � 1)(x7 � 6) Product of Powers

� 5x13 Simplify.

b. (4ab6)(�7a2b3)

(4ab6)(�7a2b3) � (4)(�7)(a � a2)(b6 � b3) Commutative and Associative Properties

� �28(a1 � 2)(b6 � 3) Product of Powers

� �28a3b9 Simplify.

POWERS OF MONOMIALS You can also look for a pattern to discover theproperty for finding the power of a power.

(42)5 � (42)(42)(42)(42)(42) (z8)3 � (z8)(z8)(z8)

� 42 � 2 � 2 � 2 � 2 � z8 � 8 � 8

� 410 � z24

Therefore, (42)5 � 410 and (z8)3 � z24. These and other similar examples suggest theproperty for finding the power of a power.

5 factors 4 factors3 factors 2 factors

23 � 25 � 2 � 2 � 2 � 2 � 2 � 2 � 2 � 2 or 28 32 � 34 � 3 � 3 � 3 � 3 � 3 � 3 or 36

These and other similar examples suggest the property for multiplying powers.

�

� ��

�

2 � 4 or 6 factors3 � 5 or 8 factors

5 factors 3 factors

Apply rule forProduct of Powers.

← ←

Power of a PowerSimplify ((32)3)2.

((32)3)2 � (32 � 3)2 Power of a Power

� (36)2 Simplify.

� 36 � 2 Power of a Power

� 312 or 531,441 Simplify.

Example 3Example 3

� � �

Lesson 8-1 Multiplying Monomials 411

2 Teach2 Teach

11

22

In-Class ExamplesIn-Class Examples

Building on PriorKnowledge

In Chapter 1, students learnedthat an algebraic expression is anexpression consisting of one ormore numbers and variablesalong with one or more arithmeticoperations. All monomials areexpressions, but not allexpressions are monomials.

MULTIPLY MONOMIALS

Determine whether eachexpression is a monomial.Explain your reasoning.

a. 17 � sThis is not a monomial becauseit involves subtraction, notmultiplication.

b. 8f 2gThis is a monomial because it isthe product of a number and twovariables.

c.

This is a monomial because it isa real number.

d. xyThis is a monomial because it isthe product of two variables.

Teaching Tip Make sure stu-dents read the Study Tip in themargin next to Example 2. Inorder for students to find theproducts of powers correctly,they must remember that theexpression x is understood tomean x1. Suggest that studentsrewrite variables without expo-nents with an exponent of 1.

Simplify each expression.

a. (r4)(�12r7) �12r11

b. (6cd5)(5c5d2) 30c6d7

3�4

PowerPoint®

Patricia Lund Divide County H.S., Crosby, ND

"The paragraph at the top of page 410 interests students because many ofthem are preparing to drive. I like to graph this data on a coordinate plane sowe review ordered pairs and graphing points. We analyze the graph to deter-mine whether or not it is linear and if it is a function. We also review functionnotation to name the graph suggested by these points."

Teacher to TeacherTeacher to Teacher


33

44

55

In-Class ExamplesIn-Class ExamplesPOWERS OF MONOMIALS

Simplify [(23)3]2. 218 or 262,144

GEOMETRY Find the volumeof a cube with a side length s � 5xyz. (5xyz)3 � 125x3y3z3

Teaching Tip Remind studentsthat when multiple sets ofgrouping symbols are used in anexpression, the outermost set isusually a pair of brackets [ ]. Theexpression [(xy)2]4 is the sameas ((xy)2)4.

Simplify [(8g3h4)2]2(2gh5)4.

65,536g16h36

Concept CheckSimplifying Expressions Janice

simplified (x2y5)3[2(xy)7] into

x13y22. Explain whether her

simplification is complete.The simplification is not completebecause the fraction is not insimplest form.

Answers

2a. (5m)2 � 25m2

2b. The power of a product is theproduct of the powers.

2c. (�3a)2 � 9a2

2d. 2(c7)3 � 2c21

3. When finding the product of powerswith the same base, keep thesame base and add the exponents.Do not multiply the bases.

4. 5 � 7d shows subtraction, notmultiplication.

5. shows division, not

multiplication.

6. A single variable is a monomial.

4a�3b

8�6

4�6

Look for a pattern in the examples below.

(xy)4 � (xy)(xy)(xy)(xy) (6ab)3 � (6ab)(6ab)(6ab)

� (x � x � x � x)(y � y � y � y) � (6 � 6 � 6)(a � a � a)(b � b � b)

� x4y4 � 63a3b3 or 216a3b3

These and other similar examples suggest the following property for finding thepower of a product.


Simplifying Monomial Expressions

Power of a Product• Words To find the power of a product, find the power of each factor

and multiply.

• Symbols For all numbers a and b and any integer m, (ab)m � ambm.

• Example (�2xy)3 � (�2)3x3y3 or �8x3y3

Power of a ProductGEOMETRY Express the area of the squareas a monomial.

Area � s2 Formula for the area of a square

� (4ab)2 s � 4ab

� 42a2b2 Power of a Product

� 16a2b2 Simplify.

The area of the square is 16a2b2 square units.

4ab

4ab

Example 4Example 4

To simplify an expression involving monomials, write an equivalent expression in which:

• each base appears exactly once,

• there are no powers of powers, and

• all fractions are in simplest form.

Simplify ExpressionsSimplify ��

13

�xy4�2[(�6y)2]3.

��13

�xy4�2

[(�6y)2]3 � ��13

�xy4�2

(�6y)6 Power of a Power

� ��13

��2x2(y4)2(�6)6y6 Power of a Product

� �19

�x2y8(46,656)y6 Power of a Power

� �19

�(46,656)x2 � y8 � y6 Commutative Property

� 5184x2y14 Product of Powers

Example 5Example 5

The properties can be used in combination to simplify more complex expressionsinvolving exponents.

Study TipPowers ofMonomialsSometimes the rules forthe Power of a Power and the Power of aProduct are combined into one rule. (ambn)p � ampbnp


Logical Give students a term such as 144a10b8 and challenge them towrite 20 unique combinations of monomials that would produce thisproduct if multiplied.

Differentiated Instruction

PowerPoint®

Lesson 8-1 Multiplying Monomials 413www.algebra1.com/self_check_quiz

GUIDED PRACTICE KEYExercises Examples

4–6 17–12 2, 3, 513, 14 4

Practice and ApplyPractice and Apply

indicates increased difficulty�

1. OPEN ENDED Give an example of an expression that can be simplified usingeach property. Then simplify each expression.

a. Product of Powers b. Power of a Power c. Power of a Product

2. Determine whether each pair of monomials is equivalent. Explain.

a. 5m2 and (5m)2 no b. (yz)4 and y4z4 yesc. �3a2 and (�3a)2 no d. 2(c7)3 and 8c21 no

3. FIND THE ERROR Nathan and Poloma are simplifying (52)(59).

Who is correct? Explain your reasoning. Poloma; see margin for explanation.

Determine whether each expression is a monomial. Write yes or no. Explain.

4. 5 � 7d no 5. �34ba� no 6. n yes

Simplify.

7. x(x4)(x6) x11 8. (4a4b)(9a2b3) 36a6b4 9. [(23)2]3 218 or 262,14410. (3y5z)2 9y10z2 11. (�4mn2)(12m2n) 12. (�2v3w4)3(�3vw3)2

�48m3n3 �72v11w18

GEOMETRY Express the area of each triangle as a monomial.

13. 5n5 14. 6a5b6

3a4b

4ab5

5n3

2n2


15. 12 yes 16. 4x3 yes 17. a � 2b no18. 4n � 5m no 19. �

yx2� no 20. �

15

�abc14 yes

Simplify.

21. (ab4)(ab2) a2b6 22. (p5q4)(p2q) p7q5

23. (�7c3d4)(4cd3) �28c4d 7 24. (�3j7k5)(�8jk8) 24j8k13

25. (5a2b3c4)(6a3b4c2) 30a5b7c6 26. (10xy5z3)(3x4y6z3) 30x5y11z6

27. (9pq7)2 81p2q14 28. (7b3c6)3 343b9c18

29. [(32)4]2 316 or 43,046,721 30. [(42)3]2 412 or 16,777,21631. (0.5x3)2 0.25x6 32. (0.4h5)3 0.064h15

33. ��34

�c�3

��2674�c3 34. ��

45

�a2�2

�12

65�a4

35. (4cd)2(�3d2)3 �432c2d8 36. (�2x5)3(�5xy6)2 �200x17y12

37. (2ag2)4(3a2g3)2 144a8g14 38. (2m2n3)3(3m3n)4 648m18n13

39. (8y3)(�3x2y2)��38

�xy4� �9x3y9 40. ��47

�m�2(49m)(17p)��

314�p5� 8m3p6�

Concept Check

Guided Practice

Application

1a–c. Sample answersare given.1a. n2(n5) � n7

1b. (n2)5 � n10

1c. (nm2)5 � n5n10

2a–d. See margin forexplanations.

Poloma

(52 ) (59) = 52 + 9

= 5 1 1

Nathan

(52)(59) = (5 . 5)2 + 9

= 2511

�

4–6. See margin for explanations.

15–20. See margin for explanations.

Homework HelpFor See

Exercises Examples15–20 121–48 2, 3, 549–54 4

Extra PracticeSee page 837.


3 Practice/Apply3 Practice/Apply

Study NotebookStudy Notebook

About the Exercises…Organization by Objective• Multiply Monomials: 15–26• Powers of Monomials: 27–54

Odd/Even AssignmentsExercises 15–42 are structuredso that students practice thesame concepts whether theyare assigned odd or evenproblems.

Assignment GuideBasic: 15–37 odd, 43–47 odd,49–52, 55–82

Average: 15–47 odd, 51–82

Advanced: 16–50 even, 53–74(optional: 75–82)

Have students—• add the definitions/examples of

the vocabulary terms to theirVocabulary Builder worksheets forChapter 8.

• include any other item(s) that theyfind helpful in mastering the skillsin this lesson.

FIND THE ERRORBoth Nathan

and Poloma added theexponents 2 and 9, which is thecorrect procedure when finding aproduct of powers. However, tellstudents to notice how Nathanand Poloma handled the bases.Suggest that students substitute avariable for 5 and then examinewhich method is correct.

Answers15. 12 is a real number and therefore a

monomial.

16. 4x3 is the product of a number and threevariables.

17. a � 2b shows subtraction, notmultiplication of variables.

18. 4n � 5m shows addition, notmultiplication of variables.

19. shows division, not

multiplication of variables.

20. �15

�abc14 is the product of a

number, �15

�, and several variables.

x�y2


Study Guide and Intervention


NAME ______________________________________________ DATE ____________ PERIOD _____

6-18-1

Less

on

8-1

Multiply Monomials A monomial is a number, a variable, or a product of a numberand one or more variables. An expression of the form xn is called a power and representsthe product you obtain when x is used as a factor n times. To multiply two powers that havethe same base, add the exponents.

Product of Powers For any number a and all integers m and n, am � an � am � n.

Simplify (3x6)(5x2).

(3x6)(5x2) � (3)(5)(x6 � x2) Associative Property

� (3 � 5)(x6 � 2) Product of Powers

� 15x8 Simplify.

The product is 15x8.

Simplify (�4a3b)(3a2b5).

(�4a3b)(3a2b5) � (�4)(3)(a3 � a2)(b � b5)� �12(a3 � 2)(b1 � 5)� �12a5b6

The product is �12a5b6.

Example 1Example 1 Example 2Example 2

ExercisesExercises

Simplify.

1. y( y5) 2. n2 � n7 3. (�7x2)(x4)

y6 n9 �7x6

4. x(x2)(x4) 5. m � m5 6. (�x3)(�x4)

x7 m6 x7

7. (2a2)(8a) 8. (rs)(rs3)(s2) 9. (x2y)(4xy3)

16a3 r2s6 4x3y 4

10. (2a3b)(6b3) 11. (�4x3)(�5x7) 12. (�3j2k4)(2jk6)

4a3b4 20x10 �6j3k10

13. (5a2bc3)� abc4� 14. (�5xy)(4x2)( y4) 15. (10x3yz2)(�2xy5z)

a3b2c7 �20x3y5 �20x4y6z3

1�5

1�3

Study Guide and Intervention, p. 455 (shown) and p. 456


1. No; this involves the quotient, not the product, of variables.

2. Yes; this is the product of a number, , and two variables.

Simplify.

3. (�5x2y)(3x4) �15x6y 4. (2ab2c2)(4a3b2c2) 8a4b4c4

5. (3cd4)(�2c2) �6c3d4 6. (4g3h)(�2g5) �8g8h

7. (�15xy4)�� xy3� 5x2y7 8. (�xy)3(xz) �x4y3z

9. (�18m2n)2�� mn2� �54m5n4 10. (0.2a2b3)2 0.04a4b6

11. � p�2p2 12. � cd3�2

c2d 6

13. (0.4k3)3 0.064k9 14. [(42)2]2 48 or 65,536

GEOMETRY Express the area of each figure as a monomial.

15. 16. 17.

18a3b6 (25x 6)� 12a3c4

GEOMETRY Express the volume of each solid as a monomial.

18. 19. 20.

27h6 m4n5 (63g4)�

21. COUNTING A panel of four light switches can be set in 24 ways. A panel of five lightswitches can set in twice this many ways. In how many ways can five light switches be set? 25 or 32

22. HOBBIES Tawa wants to increase her rock collection by a power of three this year andthen increase it again by a power of two next year. If she has 2 rocks now, how manyrocks will she have after the second year? 26 or 64

7g2

3g

m3nmn3

n

3h23h2

3h2

6ac3

4a2c

5x3

6a2b4

3ab2

1�16

1�4

4�9

2�3

1�6

1�3

1�2

b3c2�2

21a2�7b

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1Skills Practice, p. 457 and Practice, p. 458 (shown)

Reading to Learn Mathematics


NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

Pre-Activity Why does doubling speed quadruple braking distance?

Read the introduction to Lesson 8-1 at the top of page 410 in your textbook.

Find two examples in the table to verify the statement that when speed isdoubled, the braking distance is quadrupled. Write your examples in thetable.

Speed Braking Distance Speed Doubled Braking Distance (miles per hour) (feet) (miles per hour) Quadrupled (feet)

20 20 40 80

30 45 60 180

Reading the Lesson

1. Describe the expression 3xy using the terms monomial, constant, variable, and product.

The monomial 3xy is the product of the constant 3 and the variables xand y.

2. Complete the chart by choosing the property that can be used to simplify eachexpression. Then simplify the expression.

Expression Property Expression Simplified

Product of Powers

35 � 32 Power of a Power 37 or 2187Power of a Product

Product of Powers

(a3)4 Power of a Power a12

Power of a Product

Product of Powers

(�4xy)5 Power of a Power �1024x5y5

Power of a Product

Helping You Remember

3. Write an example of each of the three properties of powers discussed in this lesson.Then, using the examples, explain how the property is used to simplify them.

Sample answer: For z2 � z 5, since the bases are the same, use theProduct of Powers Property and add the exponents to get z7. For (a4)3,use the Power of a Power Property. Multiply the exponents to get a12.For (3rs)3, use the Power of a Product Property. Raise the constant andeach variable to the power to get 27r 3s3.

Reading to Learn Mathematics, p. 459

An Wang An Wang (1920–1990) was an Asian-American who became one of thepioneers of the computer industry in the United States. He grew up inShanghai, China, but came to the United States to further his studiesin science. In 1948, he invented a magnetic pulse controlling devicethat vastly increased the storage capacity of computers. He laterfounded his own company, Wang Laboratories, and became a leader inthe development of desktop calculators and word processing systems.In 1988, Wang was elected to the National Inventors Hall of Fame.

Digital computers store information as numbers. Because theelectronic circuits of a computer can exist in only one of two states,open or closed, the numbers that are stored can consist of only twodigits, 0 or 1. Numbers written using only these two digits are calledbinary numbers. To find the decimal value of a binary number, youuse the digits to write a polynomial in 2. For instance, this is how tofind the decimal value of the number 10011012. (The subscript 2 indicates that this is a binary number.)

10011012 � 1 � 2�

6�

� 0 � 2�

5�

� 0 � 2�

4�

� 1 � 2�

3�

� 1 � 2�

2�

� 0 � 2�

1�

� 1 � 2�

0�

� 1 � 6�4�

� 0 � 3�2�

� 0 � 1�6�

� 1 � 8�

� 1 � 4�

� 0 � 2�

� 1 � 1�

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1Enrichment, p. 460


41. Simplify the expression (�2b3)4 � 3(�2b4)3. 40b12

42. Simplify the expression 2(�5y3)2 � (�3y3)3. 50y 6 � 27y9

GEOMETRY Express the area of each figure as a monomial.

43. 44. 45.

GEOMETRY Express the volume of each solid as a monomial.

46. 47. 48.

TELEPHONES For Exercises 49 and 50, use the following information.The first transatlantic telephone cable has 51 amplifiers along its length. Eachamplifier strengthens the signal on the cable 106 times.

49. After it passes through the second amplifier, the signal has been boosted

106 � 106 times. Simplify this expression. 1012 or 1 trillion50. Represent the number of times the signal has been boosted after it has

passed through the first four amplifiers as a power of 106. Then simplify the expression. (106)4 or 1024

DEMOLITION DERBY For Exercises 51 and 52, use the following information. When a car hits an object, the damage is measured by the collision impact. For acertain car, the collision impact I is given by I � 2s2, where s represents the speed in kilometers per minute.

51. What is the collision impact if the speed of the car is 1 kilometer per minute? 2 kilometers per minute? 4 kilometers per minute? 2; 8; 32

52. As the speed doubles, explain what happens to the collision impact.The collision impact quadruples, since 2(2s)2 is 4(2s2).

TEST TAKING For Exercises 53 and 54, use the following information.A history test covers two chapters. There are 212 ways to answer the 12 true-falsequestions on the first chapter and 210 ways to answer the 10 true-false questions onthe second chapter.

53. How many ways are there to answer all 22 questions on the test? (Hint: Find the product of 212 and 210.) 222 or 4,194,304 ways

54. If a student guesses on each question, what is the probability of answering allquestions correctly? �

4,1914,304�

CRITICAL THINKING Determine whether each statement is true or false. If true,explain your reasoning. If false, give a counterexample.

55. For any real number a, (�a)2 � �a2. false56. For all real numbers a and b, and all integers m, n, and p, (ambn)p � ampbnp. true57. For all real numbers a, b, and all integers n, (a � b)n � an � bn. false

4n 3

2n

x 2y

y

x y 3

4k 34k 3

4k 3

7x 4

a 2b

a 2b

5f 4g3

3fg2

Demolition DerbyIn a demolition derby, thewinner is not the car thatfinishes first but the last carstill moving under its ownpower. Source: Smithsonian Magazine

15f 5g5 a4b2 (49x8)�

64k9 x3y 5 16�n5

�

�

55–57. See marginfor explanations orcounterexamples.


ELL

Open-Ended Assessment

Modeling Write an expression

such as (2x3y)3(3x2y4)2 on thechalkboard, but write the expo-nents on self-adhesive notes. Askstudent volunteers to simplifythe expression, using more self-adhesive notes to show themultiplication and addition ofexponents in each step of thesimplification process.

Getting Ready for Lesson 8-2PREREQUISITE SKILL Studentswill learn about dividingmonomials in Lesson 8-2. Part ofdividing monomials involvessimplifying fractions formed bydividing coefficients. UseExercises 75–82 to determineyour students’ familiarity withsimplifying fractions.

Answers

58. Answers should include thefollowing.

• the ratio , which

simplifies to a ratio of 1 to 4

• If s is replaced by 2s in the for-mula for the breaking distancerequired for a car traveling s miles per hour the result is

(2s)2. Using the Power of a

Product and Power of a PowerProperties, this simplifies to

4 � � s2�. This means that

doubling the speed of carmultiplies the breaking distanceby 4.

1�20

1�20

80 feet�320 feet

Maintain Your SkillsMaintain Your Skills

61–63. See pp.471A–471B.

67–70. See marginfor graphs.

Mixed Review Solve each system of inequalities by graphing. (Lesson 7-5)

61. y � 2x � 2 62. y x � 2 63. x �2y �x � 1 y � 2x � 1 y � x � 3

Use elimination to solve each system of equations. (Lesson 7-4)

64. �4x � 5y � 2 65. 3x � 4y � �25 66. x � y � 20 (4, 16)x � 2y � 6 (2, 2) 2x � 3y � 6 (�3, �4) 0.4x � 0.15y � 4

Solve each compound inequality. Then graph the solution set. (Lesson 6-4)

67. 4 � h � �3 or 4 � h 5 68. 4 � 4a � 12 � 24 {a |�2 � a � 3}69. 14 � 3h � 2 � 2 70. 2m � 3 7 or 2m � 7 9 {m |m 1}

Determine whether each transformation is a reflection, translation, dilation, or rotation. (Lesson 4-2)

71. 72. 73. reflection

dilation rotation

74. TRANSPORTATION Two trains leave York at the same time, one travelingnorth, the other south. The northbound train travels at 40 miles per hour andthe southbound at 30 miles per hour. In how many hours will the trains be 245 miles apart? (Lesson 3-7) 3�1

2� h

PREREQUISITE SKILL Simplify. (To review simplifying fractions, see pages 798 and 799.)

75. �26

� �13

� 76. �135� �

15

� 77. �150� 2 78. �

297� 3

79. �1346� �

178� 80. �

498� �

136� 81. �

4342� �

181� 82. �

4158� �

52

�

{h |h � �7 or h � 1}


58. Answer the question that was posed at the beginning ofthe lesson. See margin.

Why does doubling speed quadruple braking distance?

Include the following in your answer:

• the ratio of the braking distance required for a speed of 40 miles per hour andthe braking distance required for a speed of 80 miles per hour, and

• a comparison of the expressions �210�s2 and �

210�(2s)2.

59. 42 � 45 � ? D167 87 410 47

60. Which of the following expressions represents the volume of the cube? D

15x3 25x2

25x3 125x3DC

BA

5x

DCBA

WRITING IN MATH

Getting Ready forthe Next Lesson



4 Assess4 Assess

Answers

55. Let a � 2 and b � 3. Then (ab)2 � (2 � 3)2 or 36 and ab2 � (2)(3)2 or 18.

56. (ambn)p � (am)p(bm)p Power of a Product� ampbmp Power of a Power

57. Let a � 3, b � 4, and n � 2. Then (a � b)n � (3 � 4)2 or 49 and an � bn � 32 � 42 or 25.

67.

68.

69.

70.�4 2 4 60�2

�4 2 4 60�2

�4 2 4 60�2

�4 20�2�6�8


AlgebraActivity

Getting StartedGetting Started

TeachTeach

AssessAssess

A Follow-Up of Lesson 8-1


A Follow-Up of Lesson 8-1

Collect the Data• Cut out the pattern shown from a sheet of centimeter grid

paper. Fold along the dashed lines and tape the edges together to form a rectangular prism with dimensions 2 centimeters by 5 centimeters by 3 centimeters.

• Find the surface area SA of the prism by counting the squares on all the faces of the prism or by using the formula SA � 2w� � 2wh � 2�h, where w is the width, � is the length, and h is the height of the prism. 62 cm2

• Find the volume V of the prism by using the formula V � �wh. 30 cm3

• Now construct another prism with dimensions that are 2 times each of thedimensions of the first prism, or 4 centimeters by 10 centimeters by 6 centimeters.

• Finally, construct a third prism with dimensions that are 3 times each of thedimensions of the first prism, or 6 centimeters by 15 centimeters by 9 centimeters.

Analyze the Data1. Copy and complete the table using the prisms you made.

2. Make a prism with different dimensions from any in this activity. Repeat the stepsin Collect the Data, and make a table similar to the one in Exercise 1.

Make a Conjecture3. Suppose you multiply each dimension of a prism by 2. What is the ratio of the

surface area of the new prism to the surface area of the original prism? What isthe ratio of the volumes? 4; 8

4. If you multiply each dimension of a prism by 3, what is the ratio of the surface areaof the new prism to the surface area of the original? What is the ratio of the volumes?

5. Suppose you multiply each dimension of a prism by a. Make a conjecture aboutthe ratios of surface areas and volumes. a2; a3

Extend the Activity6. Repeat the steps in Collect the Data and Analyze the Data using cylinders. To

start, make a cylinder with radius 4 centimeters and height 5 centimeters. Tocompute surface area SA and volume V, use the formulas SA � 2�r2 � 2�rhand V � �r2h, where r is the radius and h is the height of the cylinder. Do theconjectures you made in Exercise 5 hold true for cylinders? Explain.

Investigating Surface Area and Volume3 cm

2 cm

5 cm

Prism DimensionsSurface

VolumeSurface Area Ratio Volume Ratio

Area(cm3) ��SA

SAof

oOfrNigeiwnal

�� VVof

oOfrNigeiwnal

��(cm2)

Original 2 by 5 by 3 62 30

A 4 by 10 by 6 248 240 �26428

� � 4 �23400

� � 8

B 6 by 15 by 9 558 810 �56528

� � 9 �83100

� � 27

9; 27

See pp. 471A–471B.

See pp. 471A–471B.


Teaching Algebra withManipulatives• p. 2 (master for centimeter

grid paper)• p. 135 (student recording sheet)

Glencoe Mathematics Classroom Manipulative Kit• scissors


Objective Determine surfacearea ratio and volume ratio ofrectangular prisms whendimensions are multiplied by a.

Materialscentimeter grid paperscissorstape

• Any size grid paper will workfor this activity because theunits are not important.

• Before students begin thisactivity, direct their attention tothe data table. Ask students tomake a conjecture about thesurface area and volume of theprism when each dimension ismultiplied by two, and bythree. Write student predictionson the chalkboard to use in adiscussion after the activity hasbeen completed.

• Have students verify theformula for the surface area of aprism by counting the squareson the surface of their prism.

Students should determine thatif the length, width, and heightof a rectangular prism are eachmultiplied by a, the resultingsurface area will be a2 times theoriginal surface area, and theresulting volume will be a3 timesthe original volume.

You may wish to have studentssummarize this activity and whatthey learned from it.


quiz or review of Lesson 8-1.


can you compare pHlevels?

Ask students:• Using the formula c � � �pH

,

what is the concentration ofhydrogen ions in a solutionwith pH 1? 0.1 moles/liter

• Using the same formula, what isthe concentration of hydrogenions in a solution with pH 2?0.01 moles/liter

• As the pH increases by 1, whathappens to the hydrogen ionconcentration? It gets 10 timesas small.

• According to the scale, am-monia has a pH of 12. Wouldthis indicate that ammonia has avery large or very small con-centration of hydrogen ions?a very small concentration

1�10

Dividing Monomials

Lesson 8-2 Dividing Monomials 417

Quotient of Powers• Words To divide two powers that have the same base, subtract the

exponents.

• Symbols For all integers m and n and any nonzero number a, �aa

m

n� � am � n.

• Example �bb

1

7

5� � b15 � 7 or b8

Quotient of PowersSimplify �aa

5

bb3

8�. Assume that a and b are not equal to zero.

�aa

5

bb3

8� � ��

aa

5��

bb

8

3�� Group powers that have the same base.

� (a5 � 1)(b8 � 3) Quotient of Powers

� a4b5 Simplify.

Example 1Example 1

Vocabulary• zero exponent• negative exponent

QUOTIENTS OF MONOMIALS In the following examples, the definition ofa power is used to find quotients of powers. Look for a pattern in the exponents.

5 factors 6 factors

�44

5

3� � �4 � 4

4��

44

��

44

� 4� � 4 � 4 or 42 �

33

6

2� � �3 � 3 �

33

��33 � 3� 3�� 3 � 3 � 3 � 3 or 34

3 factors 5 � 3 or 2 factors 2 factors 6 � 2 or 4 factors

These and other similar examples suggest the following property for dividing powers.��

��

1 1 1

1 1 1

1 1

1 1

TEACHING TIPAsk students why a and bcannot be 0.

can you compare pH levels?can you compare pH levels?

To test whether a solution is a base or an acid, chemists use a pH test. This test measures the concentration c of hydrogen ions (in moles per liter) in the solution.

c � ��110��

pH

The table gives examples of solutions with various pH levels. You can find the quotient of powers and use negative exponents to compare measures on the pH scale.

• Simplify expressions involving the quotient of monomials.

• Simplify expressions containing negative exponents.

Increasingacidity

Increasingalkalinity

Neutral

Source: U.S. Geological Survey

Battery acid

Lemon juice

Vinegar

Tomatoes

Coffee

Milk

Pure water

Baking soda

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Milk of Magnesia

Ammonia

Bleach

Lye

Lesson x-x Lesson Title 417

Chapter 8 Resource Masters• Study Guide and Intervention, pp. 461–462• Skills Practice, p. 463• Practice, p. 464• Reading to Learn Mathematics, p. 465• Enrichment, p. 466• Assessment, p. 517






Transparencies

LessonNotes

1 Focus1 Focus

11

22



Remind students that multiply-ing and dividing are inverse oropposite operations. With that inmind, ask students to make aconjecture about dividing mono-mials based on what they knowabout multiplying monomials.

QUOTIENTS OFMONOMIALS

Simplify . Assume that

x and y are not equal to zero.xy9

Teaching Tip Remind studentsnot to forget to find the powersof the constant terms of themonomials.

Simplify � �3. Assume that

e and f are not equal to zero.

64c9d 6��125e12f 21

4c3d2

�5e4f 7

x7y12

�x6y3

Zero Exponent and Negative Exponents

Use the key on a TI-83 Plus to evaluate expressions with exponents.

Think and Discuss 2a–d. They are reciprocals. 3. �15

�

1. Copy and complete the table below.

2. Describe the relationship between each pair of values.a. 24 and 2�4 b. 23 and 2�3 c. 22 and 2�2 d. 21 and 2�1

3. Make a Conjecture as to the fractional value of 5�1. Verify your conjectureusing a calculator.

4. What is the value of 50? 15. What happens when you evaluate 00? An error message appears.

Power of a Quotient• Words To find the power of a quotient, find the power of the numerator and

the power of the denominator.

• Symbols For any integer m and any real numbers a and b, b 0, ��ba

��m � �bam

m�.

• Example ��dc

��5 � �dc5

5�

In the following example, the definition of a power is used to compute the powerof a quotient. Look for a pattern in the exponents.

3 factors

3 factors 3 factors

This and other similar examples suggest the following property.

� �

�

NEGATIVE EXPONENTS A graphing calculator can be used to investigateexpressions with 0 as an exponent as well as expressions with negative exponents.

Power of a QuotientSimplify ��

23p2��

4.

��2

3

p2

��4

� �(2

3

p4

2)4

� Power of a Quotient

� �24(

3

p4

2)4

� Power of a Product

� �1

8

6

1

p8

� Power of a Power

24 23 22 21 20 2�1 2�2 2�3 2�4

16 8 4 2 1Power

Value�12

� �14

� �18

� �116�

GraphingCalculatorTo express a value as afraction, press

.ENTER

ENTER

Study Tip


Example 2Example 2

��25

��3

� ��25

��25

��25

�� 25

��

25

��

25

� or �25

3

3�


2 Teach2 Teach

Zero and Negative Exponents Make sure students understand that 24 and2�4 are reciprocals of each other.

After answering Exercise 4, ask students to write an expression involving division

that is equivalent to 00. A sample answer is . Show students that �

and remind them that division by zero is undefined.

0�0

06�06

06�06

PowerPoint®

Negative Exponent• Words For any nonzero number a and any integer n, a�n is the reciprocal of

an. In addition, the reciprocal of a�n is an.

• Symbols For any nonzero number a and any integer n, a�n � �a1n� and �

a1�n� � an.

• Examples 5�2 � �512� or �

215� �

m1�3� � m3

Lesson 8-2 Dividing Monomials 419www.algebra1.com/extra_examples

AlternativeMethodAnother way to look at the

problem of simplifying �22

4

4�

is to recall that anynonzero number divided

by itself is 1: �22

4

4� � �1166� or 1.

Study Tip

To understand why a calculator gives a value of 1 for 20, study the two methods

used to simplify �22

4

4�.

Method 1 Method 2

�22

4

4� � 24 � 4 Quotient of Powers �22

4

4� � �22

��

22

��

22

��

22

� Definition of powers

� 20 Subtract. � 1 Simplify.

Since �22

4

4� cannot have two different values, we can conclude that 20 � 1.

To investigate the meaning of a negative exponent, we can simplify expressions

like �88

2

5� in two ways.

Method 1 Method 2

�88

2

5� � 82 � 5 Quotient of Powers �88

2

5� � �8 � 8

8� 8

� 8� 8 � 8� Definition of powers

� 8�3 Subtract. � �813� Simplify.

Since �88

2

5� cannot have two different values, we can conclude that 8�3 � �813�.

TEACHING TIPAsk students to explainwhy 4a�2 is not a monomial. They shouldmake the connection thatan expression involving anegative exponent in thenumerator is not a monomial.

Zero Exponent• Words Any nonzero number raised to the zero power is 1.

• Symbols For any nonzero number a, a0 � 1.

• Example (�0.25)0 � 1

Zero Exponent Simplify each expression. Assume that x and y are not equal to zero.

a. ��38xx

5

yy7��

0

��3

8

xx

5

yy7��

0� 1 a0 � 1

b. �t3

ts0�

�t3

ts0� � �

t3(t1)� a0 � 1

� �tt

3� Simplify.

� t2 Quotient of Powers

1 1 1 1

1 1 1 1

1 1

1 1

Example 3Example 3


33

In-Class ExampleIn-Class ExampleNEGATIVE EXPONENTS

Teaching Tip Ask studentswhy the negative sign does notaffect the outcome. Studentsshould explain that any nonzeronumber raised to the zeropower is 1, and a negativenumber is a nonzero number.

Simplify each expression.Assume that m and n are notequal to zero.

a. � �01

b. nm0n3

�n2

12m8n7

�8m5n10

Naturalist When one-celledorganisms reproduce, thepopulation increases by a factor of 2. Population can be countedby multiplying by powers of 2.Use this pattern to show values of negative powers of 2.

23 22 21 20 2�1 2�2 2�3

8

2

� 2

2

� 2

2

� 2

4 2 1 ? ? ?


PowerPoint®

InteractiveChalkboard

PowerPoint®

Presentations

This CD-ROM is a customizableMicrosoft® PowerPoint®presentation that includes:• Step-by-step, dynamic solutions of

each In-Class Example from theTeacher Wraparound Edition

• Additional, Your Turn exercises foreach example

• The 5-Minute Check Transparencies• Hot links to Glencoe Online

Study Tools


44

55


Teaching Tip Point out tostudents that rewriting apolynomial as a product offractions makes applying theNegative Exponent Propertyeasier. Fractions that havenegative exponents can berewritten as their reciprocals.

Simplify each expression.Assume that no denominatoris equal to zero.

a.

b.

Teaching Tip Make surestudents realize that thediameter of the circle is thesame measure as the length ofa side of the square. It is veryimportant that studentsrecognize that the area of thesquare is (2r)2 and not 2r2.

TEST ITEM Refer to thefigure in Example 5 of theStudent Edition. Write theratio of the circumference ofthe circle to the area of thesquare in simplest form. C

A B

C D

Answer

3. A factor is moved from thenumerator of a fraction to thedenominator or vice versa only ifthe exponent of the factor is

negative; �4 � .1�4

2�r�

1

��2r

2��r

2r��

5r 8�p2q

75p3q�5

��15p5q�4r�8

y4�x 6z9

x�6

�y�4z9

An expression involving exponents is not considered simplified if the expressioncontains negative exponents.


Negative ExponentsSimplify each expression. Assume that no denominator is equal to zero.

a. �bd�

�

3c5

2�

�bd

�

�

3c5

2� � ��b

�

1

3��

c1

2��d

1�5�� Write as a product of fractions.

� ��b13��

c1

2��

d1

5�� a�n � �

a1n�

� �c2

bd3

5� Multiply fractions.

b. �2�1a

32ab�

7

4

cb�

7

5�

�2�1a

32ab

�

7

4

cb�

7

5� � ��21

3��

aa

�

2

4��

bb

7

7��c�1

5�� Group powers with the same base.

� ��71�(a�4 � 2)(b7 � 7)(c5)

Quotient of Powers and Negative Exponent Properties

� ��71�a�6b0c5 Simplify.

� ��71��

a16��(1)c5 Negative Exponent and

Zero Exponent Properties

� ��7ca

5

6� Multiply fractions.

Apply Properties of ExponentsMultiple-Choice Test Item

Read the Test Item

A ratio is a comparison of two quantities. It can be written in fraction form.

Solve the Test Item

• area of circle � �r2

length of square � diameter of circle or 2rarea of square � (2r)2

• �aarreeaaooffscqiurc

alree

� � �(�2r

r)

2

2� Substitute.

� ��4

�r2 � 2 Quotient of Powers

� ��4

�r0 or ��4

� r0 � 1

The answer is B.

CommonMisconceptionDo not confuse a negative number with a number raised to anegative power.

3�1 � �13

� �3 �13

�

Study Tip


Test-Taking TipSome problems can besolved using estimation.The area of the circle is less than the area of thesquare. Therefore, the ratioof the two areas must beless than 1. Use 3 as anapproximate value for �to determine which of thechoices is less than 1.

Write the ratio of the area of the circle to the area of the square in simplest form.

��2

� ��4

� �21��

�3

�DCBAr

Example 4Example 4

Example 5Example 5


Example 5 Remind students that most ofthe answer choices in multiple-choice testitems are answers that result from anarithmetic or other mistake when solving the

problem. For example, answer choice A in Example 5 is ��2

�, which is the answer you get if you use 2r2 as the area of the square.

PowerPoint®



Concept Check 1. Sample answer:9xy and 6xy2

2. �aa

3

bb2

5� � a3a�1b5b�2

� a3�1b5�2

� a2b3

www.algebra1.com/self_check_quiz



1. OPEN ENDED Name two monomials whose product is 54x2y3.

2. Show a method of simplifying �aa

3

bb2

5� using negative exponents instead of the

Quotient of Powers Property.

3. FIND THE ERROR Jamal and Emily are simplifying ��

x45x3�.

Who is correct? Explain your reasoning. Jamal; see margin for explanation.

Simplify. Assume that no denominator is equal to zero.

4. �77

8

2� 76 or 117,649 5. �xx

8

2

yy

1

7

2

� x6y5 6. ��27cz

3

2d

��3

�384c3

9dz

3

6�

7. y0(y5)(y�9) �y14� 8. 13�2 �

1169� 9. �

dc3g

�

�

5

8� �dg3c

8

5�

10. �1

�

0

5

pp6

qq

7

3� ��2qp

4

5� 11. �((cc4dd

�

9)

2

�)3

2� c11d12 12. �(4m

m

�3

nn5)0� �

m1n�

13. Find the ratio of the volume of the cylinder to thevolume of the sphere. C

�12

� 1

�32

� �32��DC

BAVolume ofcylinder � πr2h

x

2x

Volume ofsphere � πr34

3

Emi ly

�–x4

5x3� = �

x3

4

–5�

= �x4

–2�

= �41x2�

Jamal

�–

x4

5x3� = –4x3 – 5

= –4x–2

= �–x2

4�


14. �44

1

2

2� 410 or 1,048,576 15. �

33

1

7

3� 36 or 729 16. �

pp

7

4

nn

3

2� p3n

17. �yy

3

zz2

9

� y2z7 18. ��52ba

4

6n

��2

�25

4ba

8

1n2

2� 19. ��4

3xm5y

7

3��4�25

861xm20

2

y

8

12�

20. ��10

2aa8

3� ��

51a5� 21. �

4155bb5� �

31b4� 22. x3y0x�7 �

x14�

23. n2(p�4)(n�5) �n3

1p4� 24. 6�2 �

316� 25. 5�3 �

1125�

26. ��45

��2

�2156� 27. ��

32

��3

�287� 28. �

72a83ab

7

0cc

�

�

4

8� 4a4c4

29. �30

5hh

�

k�

2k3

14� �

6hk1

3

7� 30. �

1

�

8x2

3

xy2

4

yzz

7

� �9xy3z6 31. ��

�

19

3

yz1

0

6

z4

� �31z912�

32. �(5

(r2

�

r

2

3))

�

2

2� �

1010r2� 33. �

(

pp

�

5q

4

2

q)

�

�

3

1� �pq

� 34. ��rt

�

�

2

1t5

��0

1

35. ��b�4

2cc

�

3

2

dd�1��

01 36. ��5n

b2

�

z

2

�n3

4��

�1�5n

b2

2

z3� 37. ��2a3

�

a

2

bb�c2

�1��

�3�27

8ab

9

9c3

��


4–12 1–413 5


Exercises Examples14–21 1, 222–37 1–4



Guided Practice



Study NotebookStudy NotebookHave students—• add the definitions/examples of



FIND THE ERRORIt appears that

Emily was so busythinking about what to do withnegative exponents that she

accidentally rewrote �4 as .

Remind students that a negativenumber is different from anumber raised to a negativeexponent.

1�4

About the Exercises…Organization by Objective• Quotients of Monomials:

14–21, 38–39• Negative Exponents: 22–37,

40–45


Alert! You may wish to havestudents use calculators to aidthem in computation ofnumerical powers.

Assignment GuideBasic: 15–35 odd, 39–42,47–77

Average: 15–39 odd, 40–44,47–77

Advanced: 14–38 even, 43–71(optional : 72–77)

• Powers of Negative Numbers Students may assume that theexpression �63 means (�6)(�6)(�6). Explain that �63 means �(63).To express �6 to the third power, they must use parentheses, (�6)3.

• Zero Exponents The Zero Exponent property states that any nonzeronumber raised to the zero power is equal to 1. So, (�6)0 � 1. How-ever, �(60) � �1. The expression �(60) means the opposite of 6raised to the zero power. So, the opposite of 1 is �1.

Unlocking Misconceptions



Dividing Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

6-28-2

Less

on

8-2

Quotients of Monomials To divide two powers with the same base, subtract theexponents.

Quotient of Powers For all integers m and n and any nonzero number a, � am � n.

Power of a Quotient For any integer m and any real numbers a and b, b 0, � �m� .am

�bm

a�b

am�an

Simplify . Assume

neither a nor b is equal to zero.

� � �� Group powers with the same base.

� (a4 � 1)(b7 � 2) Quotient of Powers

� a3b5 Simplify.

The quotient is a3b5 .

b7�b2

a4�a

a4b7�ab2

a4b7�ab2 Simplify � �3

.

Assume that b is not equal to zero.

� �3� Power of a Quotient

� Power of a Product

� Power of a Power

� Quotient of Powers

The quotient is .8a9b9�27

8a9b9�27

8a9b15�27b6

23(a3)3(b5)3��

(3)3(b2)3

(2a3b5)3�

(3b2)32a3b5�

3b2

2a3b5�

3b2Example 1Example 1 Example 2Example 2

ExercisesExercises


1. 53 or 125 2. m2 3. p3n3

4. a 5. y 6. � y2

7. y 2 8. � �3 8a3b3 9. � �3 p6q6

10. � �4 16v4 11. � �4 r 4s8 12. r 4s4r7s7t2�s3r3t2

81�16

3r6s3�2r5s

2v5w3�v4w3

64�27

4p4q4�3p2q2

2a2b�a

xy6�y4x

1�7

�2y7�14y5

x5y3�x5y2

a2�a

p5n4�p2n

m6�m4

55�52



1. 84 or 4096 2. a3b3 3. y

4. mn 5. � 6. 2yz

7. � �38. � �2

9. �

10. x3( y�5)(x�8) 11. p(q�2)(r�3) 12. 12�2

13. � ��214. � ��4

15. 2rs5

16. � 17. 2c4f7 18. � �0 1

19. 20. � 21.

22. 23. 24.

25. � ��526. � ��1

27. � ��2

28. BIOLOGY A lab technician draws a sample of blood. A cubic millimeter of the bloodcontains 223 white blood cells and 225 red blood cells. What is the ratio of white bloodcells to red blood cells?

29. COUNTING The number of three-letter “words” that can be formed with the Englishalphabet is 263. The number of five-letter “words” that can be formed is 265. How manytimes more five-letter “words” can be formed than three-letter “words”? 676

1�484

9x2�4y2z6

2x3y2z�3x4yz�2

c8�7d2e4

7c�3d3�c5de�4

q10�r25

q�1r3�qr�2

a4�40b7

(2a�2b)�3��

5a2b4

j�k15

( j�1k3)�4�

j3k3m2�n2

m�2n�5��(m4n3)�1

r�27

r4�(3r)3

6t2u4�

v9�12t�1u5v�4��

2t�3uv5g8h2�

96f�2g3h5��54f�2g�5h3

x�3y5�4�3

8c3d2f4��4c�1d2f�3

3�u4

�15w0u�1��

5u3

22r3s2�11r2s�3

81�256

4�3

49�9

3�7

1�144

p�q2r3

1�x5y5

1�6c3

�4c2�24c5

36w10�49p12s6

6w5�7p6s3

64f 9g3�27h18

4f 3g�3h6

8y7z6�4y6z5

5d2�

45c2d3��4c2d

m5np�m4p

xy2�xy

a4b6�ab3

88�84

Practice (Average)

Dividing Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____



Dividing Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2

Pre-Activity How can you compare pH levels?


• In the formula c � � �pH, identify the base and the exponent.

base � , exponent � pH

• How do you think c will change as the exponent increases?

c will decrease.

Reading the Lesson

1. Explain what the statement � am � n means.

To divide two powers that have the same base, subtract the exponents.

2. To find c in the formula c � � �pH, you can find the power of the numerator, the power of

the denominator, and divide. This is an example of what property?Power of a Quotient Property

3. Use the Quotient of Powers Property to explain why 30 � 1. Sample answer:

� 1. The Quotient of Powers Property says that when you divide

two powers that have the same base, you subtract the exponents.

So � 30.

4. Consider the expression 4�3.

a. Explain why the expression 4�3 is not simplified. An expression involvingexponents is not considered simplified if the expression containsnegative exponents.

b. Define the term reciprocal. The reciprocal of a number is 1 divided by the number.

c. 4�3 is the reciprocal of what power of 4? 43

d. What is the simplified form of 4�3? or


5. Describe how you would help a friend who needs to simplify the expression .

Divide the constants and group powers with the same base to get

� �� . Use the Quotient of Powers Property to get (2)(x2�5) or (2)(x�3).

To simplify (2)(x�3), use the Negative Exponent Property to get

(2)� �, or .2�x3

1�x3

x2�x5

4�2

4x2�2x5

1�64

1�43

34�34

34�34

1�10

am�an

1�10

1�10


Patterns with PowersUse your calculator, if necessary, to complete each pattern.

a. 210 � b. 510 � c. 410 �

29 � 59 � 49 �

28 � 58 � 48 �

27 � 57 � 47 �

26 � 56 � 46 �

25 � 55 � 45 �

24 � 54 � 44 �

23 � 53 � 43 �

22 � 52 � 42 �

21 � 51 � 41 �

Study the patterns for a, b, and c above. Then answer the questions.

45216254641258256625161024312532409615,62564

16,38478,12512865,536390,625256262,1441,953,125512

1,048,5769,765,6251024

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____


38. The area of the rectangle is 39. The area of the triangle is 100a3b24x5y3 square units. Find the square units. Find the height of length of the rectangle. 3x2y units the triangle. 10ab units

SOUND For Exercises 40–42, use the following information.The intensity of sound can be measured in watts per square meter. The table givesthe watts per square meter for some common sounds.

40. How many times more intense is the sound from heavy traffic than the soundfrom normal conversation? 103 or 1000

41. What sound is 10,000 times as loud as a noisy kitchen? jet plane42. How does the intensity of a whisper compare to that of normal conversation?

�10

100�

PROBABILITY For Exercises 43 and 44, use the following information.If you toss a coin, the probability of getting heads is �

12

�. If you toss a coin 2 times,

the probability of getting heads each time is �12

� � �12

� or ��12

��2.

43. Write an expression to represent the probability of tossing a coin n times andgetting n heads.

44. Express your answer to Exercise 43 as a power of 2. 2�n

LIGHT For Exercises 45 and 46, use the table below.

45. Express the range of the wavelengths of visible light using positive exponents. Then evaluate each expression.

46. Express the range of the wavelengths of X-rays using positive exponents. Then evaluate each expression.

Watts/Square Meter Common Sounds

102

101

100

10-2

10-3

10-6

10-7

10-9

10-12

jet plane (30 m away)pain levelamplified music (2 m away)noisy kitchenheavy trafficnormal conversationaverage homesoft whisperbarely audible

20a2

8x3y2


SoundTimbre is the quality of the sound produced by a musical instrument. Sound quality is whatdistinguishes the sound ofa note played on a flutefrom the sound of thesame note played on atrumpet with the samefrequency and intensity.Source: www.school.discovery.com

Spectrum of Electromagnetic Radiation

Region Wavelength (cm)

Radio greater than 10

Microwave 101 to 10�2

Infrared 10�2 to 10�5

Visible 10�5 to 10�4

Ultraviolet 10�4 to 10�7

X-rays 10�7 to 10�9

Gamma Rays less than 10�9

43. ��12

��n

45. �1105� to �

1104� cm;

�100

1,000� to �

10,1000� cm

46. �1107� to �

1109� cm;

�10,00

10,000� to

�1,000,0

100,000� cm


ELL

http://www.school.discovery.com


Speaking Write an expressionfrom one of the Practice andApply problems on the chalk-board or overhead projector. Havetwo students come up to the frontof the classroom to simplify theexpression. One of the studentswill actually do the simplificationand the other one will explainwhat the partner is doing foreach step of the simplification.

Getting Ready for Lesson 8-3PREREQUISITE SKILL Studentswill learn about scientific nota-tion in Lesson 8-3. Calculationswith scientific notation involveproducts of powers of 10. UseExercises 72–77 to determineyour students’ familiarity withproducts of powers of 10.

Assessment Options

Quiz (Lessons 8-1 and 8-2) isavailable on p. 517 of the Chapter 8Resource Masters.

AnswersLesson 8-2 Dividing Monomials 423


Simplify. (Lesson 8-1)

54. (m3n)(mn2) m4n3 55. (3x4y3)(4x4y) 12x8y4

56. (a3x2)4 a12x8 57. (3cd5)2 9c2d10

58. [(23)2]2 212 or 4096 59. (�3ab)3(2b3)2 �108a3b9

NUTRITION For Exercises 60 and 61, use the following information. Between the ages of 11 and 18, you should get at least 1200 milligrams of calciumeach day. One ounce of mozzarella cheese has 147 milligrams of calcium, and oneounce of Swiss cheese has 219 milligrams. Suppose you wanted to eat no more than8 ounces of cheese. (Lesson 7-5)

60. Draw a graph showing the possible amounts of each type of cheese you can eat and still get your daily requirement of calcium. Let x be the amount ofmozzarella cheese and y be the amount of Swiss cheese. See margin.

61. List three possible solutions. Sample answers: 3 oz of mozzarella, 4 oz of Swiss; 4 oz of mozzarella, 3 oz of Swiss; 5 oz of mozzarella, 3 oz of Swiss

Write an equation of the line with the given slope and y-intercept. (Lesson 5-3)

62. slope: 1, y-intercept: �4 y � x � 4 63. slope: �2, y-intercept: 3

64. slope: ��13

�, y-intercept: �1 65. slope: �32

�, y-intercept: 2

Graph each equation by finding the x- and y-intercepts. (Lesson 4-5)

66. 2y � x � 10 67. 4x � y � 12 68. 2x � 7 � 3y66–68. See margin.Find each square root. If necessary, round to the nearest hundredth. (Lesson 2-7)

69. ��121� �11 70. �3.24� 1.8 71. ��52� �7.21

PREREQUISITE SKILL Simplify. (To review Products of Powers, see Lesson 8-1.)

72. 102 � 103 105 73. 10�8 � 10�5 10�13

74. 10�6 � 109 103 75. 108 � 10�1 107

76. 104 � 10�4 100 or 1 77. 10�12 � 10 10�11

Mixed Review

CRITICAL THINKING Simplify. Assume that no denominator is equal to zero.

47. an(a3) an � 3 48. (54x � 3)(52x � 1) 56x � 2

49. �cc

x

x

�

�

7

4� c11 50. �3b3b(

2

n

n

�

�

3

9

)� �b3n�

51. Answer the question that was posed at the beginning ofthe lesson. See pp. 471A–471B.

How can you compare pH levels?


• an example comparing two pH levels using the properties of exponents.

52. What is the value of �2�

22

2��

22

3

�3�? A

210 212 �1 �12

�

53. EXTENDED RESPONSE Write a convincing argument to show why 30 � 1using the following pattern. 35 � 243, 34 � 81, 33 � 27, 32 � 9, … Since each number is obtained by dividingthe previous number by 3, 31 � 3 and 30 � 1.

DCBA

WRITING IN MATH



63. y � �2x � 3

64. y � ��13

�x � 1

65. y � �32

�x � 2


4 Assess4 Assess

60. 66. 67. 68. y

xO

(3 , 0)12

(0, 2 )13

2x � 7 � 3y

�4�8 4 8

�4

�8

�12

y

xO

4x � y � 12

(0, �12)

(3, 0)

y

xO

2y � x � 10

(�10, 0)

(0, 5)

y

xO

x � y � 8

147x � 219y � 1200


ReadingMathematics


TeachTeach

AssessAssess

Before using this page, askstudents the difference between abicycle and a tricycle. As studentsanswer, write references to twoand three on the chalkboard oroverhead projector along with thewords bicycle and tricycle.

Prefixes Discuss with studentsthe meaning of each prefix. Thendivide the class into three or fourgroups and challenge each groupto brainstorm as many words asthey can that begin with mono-,bi-, tri-, and poly-. Give eachgroup five minutes and thenhave groups compare answers.

Students can use dictionaries tofind more words with theseprefixes. Encourage students torecord the spellings and defini-tions of new words they learnfrom the dictionary.

Ask students to summarize whatthey have learned about prefixes.

424 Investigating Slope-Intercept Form


You may have noticed that many prefixes used in mathematics are also used ineveryday language. You can use the everyday meaning of these prefixes to betterunderstand their mathematical meaning. The table shows two mathematical prefixesalong with their meaning and an example of an everyday word using that prefix.

Mathematical Prefixes and Everyday Prefixes

Prefix Everyday Meaning Examplemono-

bi-

tri-

poly-

monologue A continuousseries of jokes or comicstories delivered by onecomedian.

bicycle A vehicle consistingof a light frame mounted ontwo wire-spoked wheels onebehind the other and havinga seat, handlebars forsteering, brakes, and twopedals or a small motor bywhich it is driven.

trilogy A group of threedramatic or literary worksrelated in subject or theme.

polygon A closed planefigure bounded by three ormore line segments.

Source: The American Heritage Dictionary of the English Language

1. one; single; alone

1. two2. both3. both sides, parts, or

directions

1. three2. occurring at intervals of three3. occurring three times during

1. more than one; many; much

You can use your everyday understanding of prefixes to help you understandmathematical terms that use those prefixes.

Reading to Learn1. Give an example of a geometry term that uses one of these prefixes. Then

define that term. Sample answer: triangle; a three-sided polygon

2. MAKE A CONJECTURE Given your knowledge of the meaning of the wordmonomial, make a conjecture as to the meaning of each of the followingmathematical terms. See students’ work.a. binomial b. trinomial c. polynomial

3. Research the following prefixes and their meanings.

a. semi- precisely half of b. hexa- six c. octa- eight


English LanguageLearners may benefit fromwriting key concepts from thisactivity in their Study Notebooksin their native language and thenin English.

ELL




is scientific notationimportant in astronomy?

Ask students:• What would 1023 look like in

standard notation? a 1 with 23 zeros behind it

• Explain why you could saythat the ratio of the mass ofEarth to the mass of Saturn is

about . is equal to

• Solar System The mass of theSun is estimated to be about1.99 � 1030 kg. About howmany time heavier than Jupiteris the Sun? The Sun is about1000 times heavier than Jupiter.

1�100

1024�1026

1�100

Scientific Notation• Words A number is expressed in scientific notation when it is written as a

product of a factor and a power of 10. The factor must be greaterthan or equal to 1 and less than 10.

• Symbols A number in scientific notation is written as a � 10n, where 1 � a � 10and n is an integer.

SCIENTIFIC NOTATION When dealing with very large or very small numbers,keeping track of place value can be difficult. For this reason, numbers such as theseare often expressed in .scientific notation

Scientific Notation

Lesson 8-3 Scientific Notation 425

Vocabulary• scientific notation

is scientific notation important in astronomy?is scientific notation important in astronomy?

Reading MathStandard notation is theway in which you areused to seeing a numberwritten, where the decimalpoint determines the placevalue for each digit of thenumber.

Study Tip The following examples show one way of expressing a number that is written in scientific notation in its decimal or standard notation. Look for a relationshipbetween the power of 10 and the position of the decimal point in the standardnotation of the number.

6.59 � 104 � 6.59 � 10,000 4.81 � 10�6 � 4.81 � �1106�

� 4.81 � 0.000001

� 65,900 � 0.00000481

The decimal point moved The decimal point moved 4 places to the right. 6 places to the left.

These examples suggest the following rule for expressing a number written inscientific notation in standard notation.

• Express numbers in scientific notation and standard notation.

• Find products and quotients of numbers expressed in scientific notation.

Astronomers often work with very large numbers,such as the masses of planets.The mass of each planet inour solar system is given inthe table. Notice that eachvalue is written as theproduct of a number and apower of 10. These values arewritten in scientific notation.

Source: NASA

Planet

MercuryVenusEarthMarsJupiterSaturnUranusNeptunePluto

Mass (kilograms)

3.30 1023

4.87 1024

5.97 1024

6.42 1023

1.90 1027

5.69 1026

8.68 1025

1.02 1026

1.27 1022




Prerequisite Skills Workbook, pp. 33–36School-to-Career Masters, p. 15Science and Mathematics Lab Manual,

pp. 67–70

5-Minute Check Transparency 8-3Real-World Transparency 8Answer Key Transparencies




Transparencies

LessonNotes

1 Focus1 Focus

11

22


SCIENTIFIC NOTATION

Teaching Tip You may want toreview how to enter or find numbersin scientific notation on a calculator.For example, on a graphing calcula-tor, you can set it for scientific mode,enter a number in standard notation,press and it will display thedecimal with what exponent of 10 isused. Likewise, in normal mode, theycan use the 10x function to changea number in scientific notation tostandard notation.

Teaching Tip Caution studentsto count decimal place movescarefully because it is very easyto add or drop a zero whenconverting numbers fromscientific notation to standardnotation.

Express each number instandard notation.

a. 7.48 � 10�3 0.00748

b. 2.19 � 105 219,000

Express each number inscientific notation.

a. 0.000000672 6.72 10�7

b. 3,022,000,000,000 3.022 1012

ENTER

Standard to Scientific NotationUse these steps to express a number in scientific notation.

1. Move the decimal point so that it is to the right of the first nonzero digit. The result is a decimal number a.

2. Observe the number of places n and the direction in which you moved thedecimal point.

3. If the decimal point moved to the left, write as a � 10n. If the decimal point moved to the right, write as a � 10�n.

Standard to Scientific NotationExpress each number in scientific notation.

a. 30,500,000

30,500,000 → 3.0500000 � 10n Move decimal point 7 places to the left.

30,500,000 � 3.05 � 107 a � 3.05 and n � 7

b. 0.000781

0.000781 → 00007.81 � 10n Move decimal point 4 places to the right.

0.000781 � 7.81 � 10�4 a � 7.81 and n � �4

Example 2Example 2

To express a number in scientific notation, reverse the process used above.


Scientific to Standard NotationExpress each number in standard notation.

a. 2.45 108

2.45 � 108 � 245,000,000 n � 8; move decimal point 8 places to the right.

b. 3 10�5

3 � 10�5 � 0.00003 n � �5; move decimal point 5 places to the left.

Example 1Example 1

Scientific to Standard NotationUse these steps to express a number of the form a � 10n in standard notation.

1. Determine whether n 0 or n � 0.

2. If n 0, move the decimal point in a to the right n places.If n � 0, move the decimal point in a to the left n places.

3. Add zeros, decimal point, and/or commas as needed to indicate place value.

You will often see large numbers in the media written using a combination of anumber and a word, such as 3.2 million. To write this number in standard notation,rewrite the word million as 106. The exponent 6 indicates that the decimal pointshould be moved 6 places to the right.

3.2 million � 3,200,000

Study TipScientific NotationNotice that when anumber is in scientificnotation, no more thanone digit is to the left ofthe decimal point.


InterventionStudents mayhave previouslyencounteredscientific nota-

tion in a science class. Often inscience, numbers are roundedto two decimal places whenthey are converted from stan-dard to scientific notation. Forexample, 0.0002569 would berounded to 2.57 � 10�4 when itis converted to scientific nota-tion. Make sure students donot automatically round num-bers to two decimal placeswhen they convert to scientificnotation in this lesson.

New

Kinesthetic Have students write each digit of the number they aretrying to convert on a note card. Also have them make several zerocards to use as placeholders. Using a penny as the decimal point,students can physically move the decimal point and count the numberof places it moved.


2 Teach2 Teach

PowerPoint®

Use Scientific NotationThe graph shows chocolate andcandy sales during a recentholiday season.

a. Express the sales of candycanes, chocolates, and allcandy in standard notation.

Candy canes: $120 million � $120,000,000

Chocolates:$300 million � $300,000,000

All candy:$1.45 billion � $1,450,000,000

b. Write each of these salesfigures in scientific notation.

Candy canes:$120,000,000 � $1.2 � 108

Chocolates: $300,000,000 � $3.0 � 108

All candy: $1,450,000,000 � $1.45 � 109


PRODUCTS AND QUOTIENTS WITH SCIENTIFIC NOTATIONYou can use scientific notation to simplify computation with very large and/or very small numbers.

www.algebra1.com/extra_examples

Example 3Example 3

• Updated data• More activities on

scientific notationwww.algebra1.com/usa_today

Log on for:Log on for: A sweet holiday season

USA TODAY Snapshots®

By Marcy E. Mullins, USA TODAY

Chocolate and candy ringup holiday sales.

Source: Nielson Marketing research

Candycanes Chocolates All candy

$120million

$300million

$1.45billion

Multiplication with Scientific NotationEvaluate (5 10�8)(2.9 102). Express the result in scientific and standardnotation.

(5 � 10�8)(2.9 � 102)

� (5 � 2.9)(10�8 � 102) Commutative and Associative Properties

� 14.5 � 10�6 Product of Powers

� (1.45 � 101) � 10�6 14.5 � 1.45 � 101

� 1.45 � (101 � 10�6) Associative Property

� 1.45 � 10�5 or 0.0000145 Product of Powers

Example 4Example 4

Division with Scientific NotationEvaluate �1.

52.72829

101

509

�. Express the result in scientific and standard notation.

�1.52.72829��

11005

9� � ��1.

52.72829

��1100

9

5�� Associative Property

� 0.245 � 104 Quotient of Powers

� (2.45 � 10�1) � 104 0.245 � 2.45 � 10�1

� 2.45 � (10�1 � 104) Associative Property

� 2.45 � 103 or 2450 Product of Powers

Example 5Example 5

TEACHING TIPA billion in the U.S. andFrance means 109. A billion in Great Britainand Germany means 1012.


33

In-Class ExampleIn-Class Example

44

55


The Sporting Goods Manu-facturers Association reportedthat in 2000, women spent$4.4 billion on 124 millionpairs of shoes. Men spent$8.3 billion on 169 millionpairs of shoes.

a. Express the numbers of pairsof shoes sold to women, pairssold to men, and total spentby both men and women instandard notation. Women:124,000,000; Men: 169,000,000;total spent: $12,700,000,000.

b. Write each of these figures inscientific notation. Women:1.24 108; Men: 1.69 108;total spent: $1.27 1010.

PRODUCTS ANDQUOTIENTS WITHSCIENTIFIC NOTATION


In Lessons 8-1 and 8-2 studentslearned how to multiply anddivide monomials. Tell studentsthat if they think of numbers inscientific notation as monomials,then the procedures for multiply-ing and dividing are the same.For example, you can think of

(2.2 � 105)(3 � 10�2) as

(2.2y5)(3y�2). In this case, youfirst multiply the constants, thenthe exponents. Similarly, withscientific notation, you multiplythe constants, then the powers of 10.

Evaluate (7 � 10�6)(4.3 � 1012).Express the result in scientificand standard notation.3.01 107 and 30,100,000

Evaluate . Express

the result in scientific andstandard notation. 4 10�3

and 0.004

6.4 � 104

��1.6 � 107

Online Lesson Plans

USA TODAY Education’s Online site offers resources andinteractive features connected to each day’s newspaper.Experience TODAY, USA TODAY’s daily lesson plan, isavailable on the site and delivered daily to subscribers.This plan provides instruction for integrating USA TODAYgraphics and key editorial features into your mathematicsclassroom. Log on to www.education.usatoday.com.

PowerPoint®

PowerPoint®

http://www.education.usatoday.com


http://www.algebra1.com/usa_today




• include examples that show how tomove the decimal point whenwriting numbers in scientificnotation and vice versa.


1. Explain how you know to use a positive or a negative exponent when writing anumber in scientific notation.

2. State whether 65.2 � 103 is in scientific notation. Explain your reasoning.

3. OPEN ENDED Give an example of a large number written using a decimalnumber and a word. Write this number in standard and then in scientificnotation. Sample answer: 6.5 million; 6,500,000; 6.5 106

Express each number in standard notation.

4. 2 � 10�8 0.00000002 5. 4.59 � 103 45906. 7.183 � 1014 718,300,000,000,000 7. 3.6 � 10�5 0.000036

Express each number in scientific notation.

8. 56,700,000 5.67 107 9. 0.00567 5.67 10�3

10. 0.00000000004 4 10�11 11. 3,002,000,000,000,0003.002 1015

Evaluate. Express each result in scientific and standard notation.

12. (5.3 � 102)(4.1 � 105) 13. (2 � 10�5)(9.4 � 10�3)

14. �21..55

��

11001

2

2� 15. �21..525

��

101

�04

6�

CREDIT CARDS For Exercises 16 and 17, use the following information. During the year 2000, 1.65 billion credit cards were in use in the United States. During that same year, $1.54 trillion was charged to these cards.(Hint: 1 trillion � 1 � 1012) Source: U.S. Department of Commerce

16. Express each of these values in standard and then in scientific notation.

17. Find the average amount charged per credit card. $933.33


Concept Check1–2. See margin.


4–7 18–11 2

12–15 3, 416, 17 5




18. 5 � 10�6 0.000005 19. 6.1 � 10�9 0.000000006120. 7.9 � 104 79,000 21. 8 � 107 80,000,00022. 1.243 � 10�7 0.0000001243 23. 2.99 � 10�1 0.29924. 4.782 � 1013 47,820,000,000,000 25. 6.89 � 100 6.89

PHYSICS Express the number in each statement in standard notation.

26. There are 2 � 1011 stars in the Andromeda Galaxy. 200,000,000,00027. The center of the moon is 2.389 � 105 miles away from the center of Earth.

28. The mass of a proton is 1.67265 � 10�27 kilograms.

29. The mass of an electron is 9.1095 � 10�31 kilograms.0.00000000000000000000000000000091095


30. 50,400,000,000 5.04 1010 31. 34,402,000 3.4402 107

32. 0.000002 2 10�6 33. 0.00090465 9.0465 10�4

34. 25.8 2.58 10 35. 380.7 3.807 102

36. 622 � 106 6.22 108 37. 87.3 � 1011 8.73 1012

38. 0.5 � 10�4 5 10�5 39. 0.0081 � 10�3 8.1 10�6

40. 94 � 10�7 9.4 10�6 41. 0.001 � 1012 1 109�

�


Exercises Examples18–29 130–43 244–55 3, 456–59 5


�

27. 238,90028.0.00000000000000000000000000167265

Application16. 1,650,000,000; 1.65 109;1,540,000,000,000;1.54 1012

12–15. See margin.

Guided Practice


About the Exercises…Organization by Objective• Scientific Notation: 18–43• Products and Quotients

with Scientific Notation:44–59


Alert! Exercises 64–67 require agraphing calculator.

Assignment GuideBasic: 19–23 odd, 27–39 odd,43–57 odd, 60–63, 68–82

Average: 19–59 odd, 60–63,68–82 (optional: 64–67)


All: Practice Quiz 1 (1–10)

Answers1. When numbers between 0 and

1 are written in scientificnotation, the exponent isnegative. If the number is notbetween 0 and 1, use apositive exponent.

2. 65.2 103 is not writtenin scientific notation. Thenumber 65.2 is greaterthan 10.

12. 2.173 108; 217,300,000

13. 1.88 10�7; 0.000000188

14. 6 10�11; 0.00000000006

15. 5 109; 5,000,000,000

50. 1.5 105; 150,000

51. 4 10�4 0.0004

52. 6.2 10�7 0.00000062

53. 2.3 10�6; 0.0000023

54. 6.5 10�5; 0.000065

55. 9.3 10�7; 0.00000093


Scientific Notation

NAME ______________________________________________ DATE ____________ PERIOD _____

6-38-3

Less

on

8-3

Scientific Notation Keeping track of place value in very large or very small numberswritten in standard form may be difficult. It is more efficient to write such numbers inscientific notation. A number is expressed in scientific notation when it is written as aproduct of two factors, one factor that is greater than or equal to 1 and less than 10 and onefactor that is a power of ten.

Scientific NotationA number is in scientific notation when it is in the form a � 10n, where 1 � a � 10 and n is an integer.

Express 3.52 104 instandard notation.

3.52 � 104 � 3.52 � 10,000� 35,200

The decimal point moved 4 places to theright.

Express 37,600,000 inscientific notation.

37,600,000 � 3.76 � 107

The decimal point moved 7 places so that itis between the 3 and the 7. Since 37,600,000 1, the exponent is positive.

Express 6.21 10�5 instandard notation.

6.21 � 10�5 � 6.21 �

� 6.21 � 0.00001� 0.0000621

The decimal point moved 5 places to the left.

Express 0.0000549 inscientific notation.

0.0000549 � 5.49 � 10�5

The decimal point moved 5 places so that itis between the 5 and the 4. Since 0.0000549 � 1, the exponent is negative.

1�105



ExercisesExercises


1. 3.65 � 105 2. 7.02 � 10�4 3. 8.003 � 108

365,000 0.000702 800,300,000

4. 7.451 � 106 5. 5.91 � 100 6. 7.99 � 10�1

7,451,000 5.91 0.799

7. 8.9354 � 1010 8. 8.1 � 10�9 9. 4 � 1015

89,354,000,000 0.0000000081 4,000,000,000,000,000


10. 0.0000456 11. 0.00001 12. 590,000,0004.56 10�5 1 10�5 5.9 108

13. 0.00000000012 14. 0.000080436 15. 0.036211.2 10�10 8.0436 10�5 3.621 10�2

16. 433 � 104 17. 0.0042 � 10�3 18. 50,000,000,0004.33 106 4.2 10�6 5 1010



1. 7.3 � 107 2. 2.9 � 103 3. 9.821 � 1012

73,000,000 2900 9,821,000,000,000

4. 3.54 � 10�1 5. 7.3642 � 104 6. 4.268 � 10�6

0.354 73,642 0.000004268

PHYSICS Express the number in each statement in standard notation.

7. An electron has a negative charge of 1.6 � 10�19 Coulomb. 0.00000000000000000016

8. In the middle layer of the sun’s atmosphere, called the chromosphere, the temperatureaverages 2.78 � 104 degrees Celsius. 27,800


9. 915,600,000,000 10. 6387 11. 845,320 12. 0.000000008149.156 1011 6.387 103 8.4532 105 8.14 10�9

13. 0.00009621 14. 0.003157 15. 30,620 16. 0.00000000001129.621 10�5 3.157 10�3 3.062 104 1.12 10�11

17. 56 � 107 18. 4740 � 105 19. 0.076 � 10�3 20. 0.0057 � 103

5.6 108 4.74 108 7.6 10�5 5.7 or 5.7 100


21. (5 � 10�2)(2.3 � 1012) 22. (2.5 � 10�3)(6 � 1015)1.15 1011; 115,000,000,000 1.5 1013; 15,000,000,000,000

23. (3.9 � 103)(4.2 � 10�11) 24. (4.6 � 10�4)(3.1 � 10�1)1.638 10�7; 0.0000001638 1.426 10�4; 0.0001426

25. 26. 27.

2.0 106; 2,000,000 1.6 10�5; 0.000016 2.34 102; 234

28. 29. 30.

2.0 10�3; 0.002 2.0 107; 20,000,000 6.5 10�6; 0.0000065

31. BIOLOGY A cubic millimeter of human blood contains about 5 � 106 red blood cells. Anadult human body may contain about 5 � 106 cubic millimeters of blood. About how manyred blood cells does such a human body contain? about 2.5 1013 or 25 trillion

32. POPULATION The population of Arizona is about 4.778 � 106 people. The land area isabout 1.14 � 105 square miles. What is the population density per square mile?about 42 people per square mile

2.015 � 10�3��

3.1 � 102

1.68 � 104��8.4 � 10�4

1.82 � 105��9.1 � 107

1.17 � 102��5 � 10�1

6.72 � 103��4.2 � 108

3.12 � 103��1.56 � 10�3

Practice (Average)

Scientific Notation

NAME ______________________________________________ DATE ____________ PERIOD _____



Scientific Notation

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3

Less

on

8-3

Pre-Activity Why is scientific notation important in astronomy?


In the table, each mass is written as the product of a number and a powerof 10. Look at the first factor in each product. How are these factors alike?

They are all greater than 1 and less than 10.

Reading the Lesson

1. Is the number 0.0543 � 104 in scientific notation? Explain.

No; the first factor is less than 1.

2. Complete each sentence to change from scientific notation to standard notation.

a. To express 3.64 � 106 in standard notation, move the decimal point

places to the .

b. To express 7.825 � 10�3 in standard notation, move the decimal point

places to the .

3. Complete each sentence to change from standard notation to scientific notation.

a. To express 0.0007865 in scientific notation, move the decimal point places

to the right and write .

b. To express 54,000,000,000 in scientific notation, move the decimal point

places to the left and write .

4. Write positive or negative to complete each sentence.

a. powers of 10 are used to express very large numbers in

scientific notation.

b. powers of 10 are used to express very small numbers in

scientific notation.


5. Describe the method you would use to estimate how many times greater the mass ofSaturn is than the mass of Pluto.

Divide 5.69 1026 by 1.27 1022. Since 5.69 � 1.27 � 4.48 and 1026 � 1022 is 104, the mass of Mars is about 4.48 104 times the mass of Pluto.

Negative

Positive

5.4 1010

107.865 10�4

4

left3

right6


Converting Metric UnitsScientific notation is convenient to use for unit conversions in the metric system.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3

How many kilometersare there in 4,300,000 meters?

Divide the measure by the number ofmeters (1000) in one kilometer. Expressboth numbers in scientific notation.

� 4.3 � 103

The answer is 4.3 3 103 km.

4.3 � 106��1 � 103

Convert 3700 grams intomilligrams.

Multiply by the number of milligrams(1000) in 1 gram.

(3.7 � 103)(1 � 103) � 3.7 � 106

There are 3.7 � 106 mg in 3700 g.


Complete the following. Express each answer in scientific notation.

1. 250,000 m � km 2. 375 km � m

3. 247 m � cm 4. 5000 m � mm5 � 1062.47 � 104

3.75 � 1052.5 � 102

Enrichment, p. 472

The distances of the

planets from the Sun can

be written in scientific

notation. Visit

www.algebra1.com/webquest to continue

work on your WebQuest

project.


42. STARS In the 1930s, the Indian physicist Subrahmanyan Chandrasekhar andothers predicted the existence of neutron stars. These stars can have a density of 10 billion tons per teaspoonful. Express this density in scientific notation.1 1010 tons

43. PHYSICAL SCIENCE The unit of measure for counting molecules is a mole. One mole of a substance is the amount that contains about602,214,299,000,000,000,000,000 molecules. Write this number in scientificnotation. 6.02214299 1023


44. (8.9 � 104)(4 � 103) 45. (3 � 106)(5.7 � 102)

46. (5 � 10�2)(8.6 � 10�3) 47. (1.2 � 10�5)(1.2 � 10�3)

48. (3.5 � 107)(6.1 � 10�8) 49. (2.8 � 10�2)(9.1 � 106)

50. �74..28

��

1100

9

4� 51. �71..28

��

1100

3

7� 52. �3.156.12

��

11002

�4�

53. �1.043.55

��

11003

�2� 54. �2.

47.935��

1100�

�

4

8� 55. �

4.655��

11005

�1�

50–55. See margin.56. HAIR GROWTH The usual growth rate of human hair is 3.3 � 10�4 meter per

day. If an individual hair grew for 10 years, how long would it be in meters?(Assume 365 days in a year.) about 1.2 m

57. NATIONAL DEBT In April 2001, the national debt was about $5.745 trillion, andthe estimated U.S. population was 283.9 million. About how much was eachU.S. citizen’s share of the national debt at that time? about $20,236

Online Research Data Update What is the current U.S. population andamount of national debt? Visit www.algebra1.com/data_update to learn more.

58. BASEBALL The table below lists the greatest yearly salary for a major leaguebaseball player for selected years.

About how many times as great was the yearly salary of Alex Rodriguez in 2000as that of George Foster in 1982? about 12 times

59. ASTRONOMY The Sun burns about 4.4 � 106 tons of hydrogen per second.How much hydrogen does the Sun burn in one year? (Hint: First, find thenumber of seconds in a year and write this number in scientific notation.)about 1.4 1014 or 140 trillion tons

60. CRITICAL THINKING Determine whether each statement is sometimes, always, ornever true. Explain your reasoning. a–b. See pp. 471A–471B.a. If 1 � a � 10 and n and p are integers, then (a � 10n)p � ap � 10np.

b. The expression ap � 10np in part a is in scientific notation.

Source: USA TODAY

Baseball Salary MilestonesYear

1979

1982

1990

1992

1996

1997

2000

Player

Nolan Ryan

George Foster

Jose Canseco

Ryne Sandberg

Ken Griffey, Jr.

Pedro Martinez

Alex Rodriguez

Yearly Salary

$1 million

$2.04 million

$4.7 million

$7.1 million

$8.5 million

$12.5 million

$25.2 million

44. 3.56 108;356,000,00045. 1.71 109;1,710,000,00046. 4.3 10�4;0.000004347. 1.44 10�8;0.000000014448. 2.135 100;2.13549. 2.548 105;254,800


�

BaseballThe contract AlexRodriguez signed with the Texas Rangers onDecember 11, 2000,guarantees him $25.2 million a year for 10 seasons. Source: Associated Press


ELL

http://www.algebra1.com/webquest


http://www.algebra1.com/data_update


Writing Have students lookthrough their science book andfind numbers expressed inscientific notation. Ask studentsto write a sentence explainingwhat each number represents inscience.

Getting Ready for Lesson 8-4PREREQUISITE SKILL Studentswill learn about polynomials inLesson 8-4. They will identify,write, and simplify polynomialsand must know how to evaluateexpressions. Use Exercises 77–82to determine your students’familiarity with evaluatingexpressions.

Assessment Options

Practice Quiz 1 The quiz pro-vides students with a brief reviewof the concepts and skills inLesson 8-1 through 8-3. Lessonnumbers are given to the right ofthe exercises or instruction linesso students can review conceptsnot yet mastered.

Answers

74.

75.

76.�36 �34 �32�38 �30

14 18 2012 22

�16 �14 �12�18 �10



GraphingCalculator

Practice Quiz 1Practice Quiz 1


1. n3(n4)(n) n8 2. 4ad(3a3d) 12a4d2 3. (�2w3z4)3(�4wz3)2 �128w11z18

Simplify. Assume that no denominator is equal to zero. (Lesson 8-2)

4. �2

1

5

5

pp

1

3

0

� �53p7� 5. ��7

6nkp

3

4��2

�4396nk2p

6

8� 6. �(3y

4�

x3

0

zy5

2

)�2� �36

yz4

10�

Evaluate. Express each result in scientific and standard notation. (Lesson 8-3) 7–10. See margin.7. (6.4 � 103)(7 � 102) 8. (4 � 102)(15 � 10�6) 9. �

92..23

��

1100

3

5� 10. �13.2.6

��

1100�

7

2�

Lessons 8-1 through 8-3


Mixed Review


Simplify. Assume no denominator is equal to zero. (Lesson 8-2)

68. �479aab

4b4c

7

3c2

� �7a

c

3b3� 69. �

�4

nn�

3p2

�5

� ��4pn5

5� 70. �

((38nn2

7

))�

2

3� 1728n20

Determine whether each expression is a monomial. Write yes or no. (Lesson 8-1)

71. 3a � 4b no 72. �n6

� no 73. �v3

2� yes

Solve each inequality. Then check your solution and graph it on a number line.(Lesson 6-1) 74–76. See margin for graphs.74. m � 3 � �17 75. �9 � d 9 76. �x � 11 23

{mm � �14} {dd 18} {xx � �34}PREREQUISITE SKILL Evaluate each expression when a � 5, b � �2, and c � 3.(To review evaluating expressions, see Lesson 1-2.)

77. 5b2 20 78. c2 � 9 0 79. b3 � 3ac 3780. a2 � 2a � 1 34 81. �2b4 � 5b3 � b 10 82. 3.2c3 � 0.5c2 � 5.2c

75.3


Why is scientific notation important in astronomy?


• the mass of each of the planets in standard notation, and

• an explanation of how scientific notation makes presenting and computingwith large numbers easier.

62. Which of the following is equivalent to 360 � 10�4? C3.6 � 103 3.6 � 102 3.6 � 10�2 3.6 � 10�3

63. SHORT RESPONSE There are an average of 25 billion red blood cells in thehuman body and about 270 million hemoglobin molecules in each red bloodcell. Find the average number of hemoglobin molecules in the human body.6.75 1018

SCIENTIFIC NOTATION You can use a graphing calculator to solve problemsinvolving scientific notation. First, put your calculator in scientific mode. To enter4.5 � 109, enter 4.5 10 9.

64. (4.5 � 109)(1.74 � 10�2) 7.83 107 65. (7.1 � 10�11)(1.2 � 105) 8.52 10�6

66. (4.095 � 105) � (3.15 � 108) 67. (6 � 10�4) � (5.5 � 10�7) 1.09 103

1.3 10�3

�

DCBA

WRITING IN MATH


4 Assess4 Assess

Answers

7. 4.48 106; 4,480,000

8. 6 10�3; 0.006

9. 4 10�2; 0.04

10. 3 109; 3,000,000,000


AlgebraActivity


TeachTeach

AssessAssess

A Preview of Lesson 8-4

Objective Use algebra tiles tomodel polynomials.

Materialsalgebra tiles

• Prior to the activity, make surestudents easily recognize eachtype of algebra tile. Remindthem that constants are repre-sented by the 1 and �1 tiles,variables without exponentsare represented by the x tiles,

and x2 tiles represent variableswith an exponent of 2.

• Make sure students understandthat the number of x and x2 tiles represent the coefficientsof x and x2, respectively. Thenumber of 1 tiles represents theconstant in the expression.

• Tell students to be careful touse the tiles with the correctsign. It is easy to accidentallysubstitute an x tile when youshould use a �x tile.

Speak a polynomial and havestudents model the polynomial.


Investigating Slope-Intercept Form 431Algebra Activity Polynomials 431


Algebra tiles can be used to model polynomials. A polynomial is a monomial or the sum of monomials. The diagram at the right shows the models.

Use algebra tiles to model each polynomial.

• 4xTo model this polynomial, you will need 4 greenx tiles.

• 2x2 � 3To model this polynomial, you will need 2 bluex2 tiles and 3 red �1 tiles.

• �x2 � 3x � 2 To model this polynomial, you will need 1 red�x2 tile, 3 green x tiles, and 2 yellow 1 tiles.

Model and AnalyzeUse algebra tiles to model each polynomial. Then draw a diagram ofyour model. 1–4. See pp. 471A–471B.1. �2x2 2. 5x � 4 3. 3x2 � x 4. x2 � 4x � 3

Write an algebraic expression for each model.5. 6.

3x2 � 2x �x2 � x � 4

7. 8.

�2x2 � 3x � 1 x2 � 2x � 3

9. MAKE A CONJECTURE Write a sentence or two explaining why algebra tiles are sometimes called area tiles. x2, x, and 1 represent the areas of the tiles.

x 2�x �x

1 1 1

x x x�x 2�x 2

�1

x1 1 1 1

�x 2x 2 x 2 x 2�x �x

x x x1 1

�x 2

�1 �1�1

x 2 x 2

x x x x

Polynomials Polynomial Models

Polynomials are modeled usingthree types of tiles.

Each tile has an opposite.

x 2x1

�x 2�1 �x

Algebra Activity Polynomials 431

Teaching Algebra withManipulatives• pp. 10–11 (master for algebra tiles)• p. 136 (student recording sheet)

Glencoe Mathematics Classroom Manipulative Kit• algebra tiles




Mathematical Background notesare available for this lesson on p. 408D.

are polynomials usefulin modeling data?

Ask students:• What is the value of t for the

year 1994? 2 What would bethe value of t for the year 1997? 5

• Using the equation, find thevalue of H for the year 1997.34

• Why does the value of H for theyear 1997 differ from the valuein the table? The polynomialmodel approximates but does notnecessarily match the real valuesexactly.

DEGREE OF A POLYNOMIAL A is a monomial or a sum ofmonomials. Some polynomials have special names. A is the sum of twomonomials, and a is the sum of three monomials. Polynomials with morethan three terms have no special names.

trinomialbinomial

polynomial

Vocabulary• polynomial • binomial• trinomial • degree of a monomial• degree of a polynomial

Polynomials


are polynomials useful in modeling data?are polynomials useful in modeling data?

Identify PolynomialsState whether each expression is a polynomial. If it is a polynomial, identify itas a monomial, binomial, or trinomial.

a.

b.

c.

d.

Example 1Example 1

Study Tip

CommonMisconceptionBefore deciding if anexpression is a polynomial,write each term of theexpression so that thereare no variables in thedenominator. Then lookfor negative exponents.Recall that the exponentsof a monomial must benonnegative integers.

Study Tip

The number of hours H spent per person per year playing video games from 1992 through 1997 is shown in the table. These data can be modeled by the equation

H � �14

�(t4 � 9t3 � 26t2 � 18t � 76),

where t is the number of years since 1992. The expression t4 � 9t3 � 26t2 � 18t � 76 is an example of a polynomial.

• Find the degree of a polynomial.

• Arrange the terms of a polynomial in ascending or descending order.

Year

199219931994199519961997

Hours spentper person

191922242636

Source: U.S. Census Bureau

VideoGameUsage

Monomial Binomial Trinomial

7 3 � 4y x � y � z

13n 2a � 3c p2 � 5p � 4

�5z3 6x2 � 3xy a2 � 2ab � b2

4ab3c 2 7pqr � pq2 3v2 � 2w � ab3

Monomial, Binomial,Expression Polynomial?

or Trinomial?

Yes, 2x � 3yz � 2x � (�3yz). The 2x � 3yz

expression is the sum of two monomials.binomial

8n3 � 5n�2 No. 5n�2 � �n52�, which is not a monomial. none of these

�8 Yes. �8 is a real number. monomial

Yes. The expression simplifies to 4a2 � 5a � a � 9 4a2 � 6a � 9, so it is the sum of three trinomial

monomials.

Like TermsBe sure to combine anylike terms before decidingif a polynomial is amonomial, binomial, ortrinomial.

LessonNotes

1 Focus1 Focus

Chapter 8 Resource Masters• Study Guide and Intervention, pp. 473–474• Skills Practice, p. 475• Practice, p. 476• Reading to Learn Mathematics, p. 477• Enrichment, p. 478• Assessment, pp. 517, 519

Graphing Calculator and Spreadsheet Masters, p. 37



TechnologyInteractive Chalkboard



Transparencies

44

In-Class ExampleIn-Class ExampleLesson 8-4 Polynomials 433

The is the sum of the exponents of all its variables.

The is the greatest degree of any term in the polynomial. To find the degree of a polynomial, you must find the degree of each term.

degree of a polynomial

degree of a monomial

WRITE POLYNOMIALS IN ORDER The terms of a polynomial are usuallyarranged so that the powers of one variable are in ascending (increasing) order ordescending (decreasing) order.


Write a PolynomialGEOMETRY Write a polynomial to represent the area of the shaded region.

Words The area of the shaded region is the area of the rectangle minus the area of the circle.

Variables area of shaded region � Awidth of rectangle � 2rrectangle area � b(2r)circle area � �r2

area of shaded region � rectangle area � circle areaEquation A � b(2r) � �r2

A � 2br � �r2

The polynomial representing the area of the shaded region is 2br � �r2.

r

b

Example 2Example 2

Degree of a PolynomialFind the degree of each polynomial.

a.b.c.

Example 3Example 3

Arrange Polynomials in Ascending Order Arrange the terms of each polynomial so that the powers of x are in ascendingorder.

a. 7x2 � 2x4 � 11

7x2 � 2x4 � 11 � 7x2 � 2x4 � 11x0 x0 � 1

� �11 � 7x2 � 2x4 Compare powers of x: 0 � 2 � 4.

b. 2xy3 � y2 � 5x3 � 3x2y2xy3 � y2 � 5x3 � 3x2y

� 2x1y3 � y2 � 5x3 � 3x2y1 x � x1

� y2 � 2xy3 � 3x2y � 5x3 Compare powers of x: 0 � 1 � 2 � 3.

Example 4Example 4

Study Tip

TEACHING TIPThe number 0 has nodegree.

Polynomials can be used to express geometric relationships.

Monomial Degree

8y 4 4

3a 1

�2xy2z 3 1 � 2 � 3 or 6

7 0

Polynomial TermsDegree of Degree of Each Term Polynomial

5mn2 5mn2 1, 2 3

�4x2y2 � 3x2 � 5 �4x2y2, 3x2, 5 4, 2, 0 4

3a � 7ab � 2a2b � 16 3a, 7ab, 2a2b, 16 1, 2, 3, 0 3

Degrees of 1 and 0• Since a � a1, the

monomial 3a can berewritten as 3a1. Thus3a has degree 1.

• Since x0 � 1, themonomial 7 can berewritten as 7x0. Thus 7 has degree 0.

Lesson 8-4 Polynomials 433

11

22

33


DEGREE OF APOLYNOMIAL

Teaching Tip Since studentsmust recall the definition of amonomial in order to define apolynomial, review the definitionof a polynomial from Lesson 8-1.Remind students that monomialsare the product of a numberand one or more variables, so

expressions such as are notmonomials.

State whether each expressionis a polynomial. If it is a poly-nomial, identify it as a mono-mial, binomial, or trinomial.

a. 6 � 4 yes, binomial

b. x2 � 2xy � 7 yes, trinomial

c. no

d. 26b5 yes, monomial

GEOMETRY Write apolynomial to represent thearea of the shaded region.

2bh � bh

Find the degree of eachpolynomial.

a. 12 � 5b � 6bc � 8bc2 3

b. 9x2 � 2x � 4 2

c. 14g2h5i 8

WRITE POLYNOMIALS IN ORDER

Arrange the terms of eachpolynomial so that the powersof x are in ascending order.

a. 16 � 14x3 � 2x � x2

16 � 2x � x2 � 14x3

b. 7y2 � 4x3 � 2xy3 � x2y2

7y2 � 2xy3 � x2y2 � 4x3

1�2

b

h

14d � 19e2

��5d4

1�b2

Degree of Polynomials Make sure students do not confuse thedegree of a polynomial with the number of terms. For example, x3 � 1is a binomial, but the degree is three, not two, because the greatestdegree of any of the terms is three.

Unlocking Misconceptions

2 Teach2 Teach

PowerPoint®

PowerPoint®




55


Arrange the terms of eachpolynomial so that the powersof x are in descending order.

a. 8 � 7x2 � 12xy3 � 4x3y�4x3y � 7x2 � 12xy3 � 8

b. a4 � ax2 � 2a3xy3 � 9x4y�9x4y � ax2 � 2a3xy3 � a4

Have students—• add the definitions/examples of



Guided Practice

Application

Arrange Polynomials in Descending OrderArrange the terms of each polynomial so that the powers of x are in descendingorder.

a. 6x2 � 5 � 8x � 2x3

6x2 � 5 � 8x � 2x3 � 6x2 � 5x0 � 8x1 � 2x3 x0 � 1 and x � x1

� �2x3 � 6x2 � 8x � 5 3 2 1 0

b. 3a3x2 � a4 � 4ax5 � 9a2x3a3x2 � a4 � 4ax5 � 9a2x � 3a3x2 � a4x0 � 4a1x5 � 9a2x1 a � a1, x0 � 1, and x � x1

� 4ax5 � 3a3x2 � 9a2x � a4 5 2 1 0

Example 5Example 5


4–6 17–9 3

10, 11 412, 13 5

14 2



1. OPEN ENDED Give an example of a monomial of degree zero.

2. Explain why a polynomial cannot contain a variable raised to a negative power.

3. Determine whether each statement is true or false. If false, give a counterexample.

a. All binomials are polynomials. trueb. All polynomials are monomials. false; 3x � 5c. All monomials are polynomials. true

State whether each expression is a polynomial. If the expression is a polynomial,identify it as a monomial, a binomial, or a trinomial.4. 5x � 3xy � 2x 5. �

25z� yes; monomial 6. 9a2 � 7a � 5

yes; binomial yes; trinomial Find the degree of each polynomial.

7. 1 0 8. 3x � 2 1 9. 2x2y3 � 6x4 5

Arrange the terms of each polynomial so that the powers of x are in ascending order.

10. 6x3 � 12 � 5x �12 � 5x � 6x3 11. �7a2x3 � 4x2 � 2ax5 � 2a2a � 4x2 � 7a2x3 � 2ax5

Arrange the terms of each polynomial so that the powers of x are in descending order.

12. 2c5 � 9cx2 � 3x 9cx2 � 3x � 2c5 13. y3 � x3 � 3x2y � 3xy2

x3 � 3x2y � 3xy2 � y3

14. GEOMETRY Write a polynomial to represent the area of the shaded region. 2cd � �d2

c

2d

Concept Check2. See margin.

State whether each expression is a polynomial. If the expression is a polynomial,identify it as a monomial, a binomial, or a trinomial.

15. 14 yes; monomial 16. �6m

p

2� � p3 no

17. 7b � 3.2c � 8b yes; binomial 18. �13

�x2 � x � 2 yes; trinomial

19. 6gh2 � 4g2h � g yes; trinomial 20. �4 � 2a � �a52� no


Exercises Examples15–20 121–24 225–36 337–52 4, 5

Sampleanswer: �8



Auditory/Musical Have students work in pairs. Give them apolynomial and a rhythm instrument (or have them tap on their desks).For each monomial have them tap out each exponent as a beat. Havethe partner record the total number of beats for each monomial. Thendecide the degree of the polynomial based on the monomial with thegreatest number of beats.


PowerPoint®

Answer

2. A variable with a negative powerwould indicate the quotient of anumber, 1, and a variable.Monomials can only be a number,variable, or the product of anumber and one or morevariables. If one of the terms of anexpression is not a monomial,then the expression is not apolynomial.


Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

6-48-4

Less

on

8-4

Degree of a Polynomial A polynomial is a monomial or a sum of monomials. Abinomial is the sum of two monomials, and a trinomial is the sum of three monomials.Polynomials with more than three terms have no special name. The degree of a monomialis the sum of the exponents of all its variables. The degree of the polynomial is the sameas the degree of the monomial term with the highest degree.

State whether each expression is a polynomial. If the expression isa polynomial, identify it as a monomial, binomial, or trinomial. Then give thedegree of the polynomial.

Expression Polynomial?Monomial, Binomial, Degree of the

or Trinomial? Polynomial

3x � 7xyzYes. 3x � 7xyz � 3x � (�7xyz),

binomial 3which is the sum of two monomials

�25 Yes. �25 is a real number. monomial 0

7n3 � 3n�4No. 3n�4 � , which is not

none of these —a monomial

Yes. The expression simplifies to9x3 � 4x � x � 4 � 2x 9x3 � 7x � 4, which is the sum trinomial 3

of three monomials

State whether each expression is a polynomial. If the expression is a polynomial,identify it as a monomial, binomial, or trinomial.

1. 36 yes; monomial 2. � 5 no

3. 7x � x � 5 yes; binomial 4. 8g2h � 7gh � 2 yes; trinomial

5. � 5y � 8 no 6. 6x � x2 yes; binomial

Find the degree of each polynomial.

7. 4x2y3z 6 8. �2abc 3 9. 15m 1

10. s � 5t 1 11. 22 0 12. 18x2 � 4yz � 10y 2

13. x4 � 6x2 � 2x3 � 10 4 14. 2x3y2 � 4xy3 5 15. �2r8s4 � 7r2s � 4r7s6 13

16. 9x2 � yz8 9 17. 8b � bc5 6 18. 4x4y � 8zx2 � 2x5 5

19. 4x2 � 1 2 20. 9abc � bc � d5 5 21. h3m � 6h4m2 � 7 6

1�4y2

3�q2

3�n4

ExampleExample

ExercisesExercises


State whether each expression is a polynomial. If the expression is a polynomial,identify it as a monomial, a binomial, or a trinomial.

1. 7a2b � 3b2 � a2b 2. y3 � y2 � 9 3. 6g2h3k

yes; binomial yes; trinomial yes; monomial

GEOMETRY Write a polynomial to represent the area of each shaded region.

4. ab � b2 5. d 2 � �d2


6. x � 3x4 � 21x2 � x3 4 7. 3g2h3 � g3h 5

8. �2x2y � 3xy3 � x2 4 9. 5n3m � 2m3 � n2m4 � n2 6

10. a3b2c � 2a5c � b3c2 6 11. 10s2t2 � 4st2 � 5s3t2 5

Arrange the terms of each polynomial so that the powers of x are in ascendingorder.

12. 8x2 � 15 � 5x5 13. 10bx � 7b2 � x4 � 4b2x3

�15 � 8x2 � 5x5 �7b2 � 10bx � 4b2x3 � x4

14. �3x3y � 8y2 � xy4 15. 7ax � 12 � 3ax3 �a2x2

8y2 � xy4 � 3x3y �12 � 7ax � a2x2 � 3ax3

Arrange the terms of each polynomial so that the powers of x are in descendingorder.

16. 13x2 � 5 � 6x3 � x 17. 4x � 2x5 � 6x3 � 26x3 � 13x2 � x � 5 2x5 � 6x3 � 4x � 2

18. g2x � 3gx3 � 7g3 � 4x2 19. �11x2y3 � 6y � 2xy �2x4

�3gx3 � 4x2 � g2x � 7g3 2x4 � 11x2y3 � 2xy � 6y

20. 7a2x2 � 17 � a3x3 � 2ax 21. 12rx3 � 9r6 � r2x � 8x6

�a3x3 � 7a2x2 � 2ax � 17 8x6 � 12rx3 � r 2x � 9r 6

22. MONEY Write a polynomial to represent the value of t ten-dollar bills, f fifty-dollarbills, and h one-hundred-dollar bills. 10t � 50f � 100h

23. GRAVITY The height above the ground of a ball thrown up with a velocity of 96 feet persecond from a height of 6 feet is 6 � 96t � 16t2 feet, where t is the time in seconds.According to this model, how high is the ball after 7 seconds? Explain.�106 ft; The height is negative because the model does not account for the ball hitting the ground when the height is 0 feet.

1�4

da

b

b

1�5

Practice (Average)

Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____



Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4

Less

on

8-4

Pre-Activity How are polynomials useful in modeling data?


• How many terms does t4 � 9t3 � 26t � 18t � 76 have?

five• What could you call a polynomial with just one term?

a monomial

Reading the Lesson

1. What is the meaning of the prefixes mono-, bi-, and tri-?

Mono- means one, bi- means two, and tri- means three.

2. Write examples of words that begin with the prefixes mono-, bi-, and tri-.

Sample answer: monocycle (one wheel), bicycle (two wheels), tricycle(three wheels)

3. Complete the table.

monomial binomial trinomialpolynomial with more

than three terms

Example 3r 2t 2x2 � 3x 5x2 � 3x � 2 7s2 � s4 � 2s3 � s � 5

Number of Terms 1 2 3 5

4. What is the degree of the monomial 3xy2z? 4

5. What is the degree of the polynomial 4x4 � 2x3y3 � y2 � 14? Explain how you foundyour answer.

6; Since 0 � 4 � 4 , 4x4 has degree 4; since 0 � 3 � 3 � 6, 2x3y3 hasdegree 6; y2 has degree 2; and 14 has degree 0. The highest degree ofthese terms is 6.


6. Use a dictionary to find the meaning of the terms ascending and descending. Write theirmeanings and then describe a situation in your everyday life that relates to them.

ascending: going, growing, or moving upward; descending: moving froma higher to a lower place; Sample answer: climbing stairs, hiking


Polynomial FunctionsSuppose a linear equation such as 23x � y � 4 is solved for y. Then an equivalent equation,y � 3x � 4, is found. Expressed in this way, y is a function of x, or f(x) � 3x � 4. Notice that the right side of the equation is a binomial of degree 1.

Higher-degree polynomials in x may also form functions. An example is f(x) � x3 � 1, which is a polynomial function of degree 3. You can graph this function using a table of ordered pairs, as shown at the right.

For each of the following polynomial functions, make a tableof values for x and y � f(x). Then draw the graph on the grid.

1. f(x) � 1 � x2 2. f(x) � x2 � 5y

x

y

O

x y

�1�12

� �2�38

�

�1 0

0 1

1 2

1�12

� 4 �38

�

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____



GEOMETRY Write a polynomial to represent the area of each shaded region.

21. 0.5bh 22. ab � 4x2

23. 24. �r2 � r2


25. 5x3 3 26. 9y 1 27. 4ab 228. �13 0 29. c4 � 7c2 4 30. 6n3 � n2p2 431. 15 � 8ag 2 32. 3a2b3c4 � 18a5c 9 33. 2x3 � 4y � 7xy 334. 3z5 � 2x2y3z � 4x2z 6 35. 7 � d5 � b2c2d3 � b6 7 36. 11r2t4 � 2s4t5 � 24 9

Arrange the terms of each polynomial so that the powers of x are in ascending order. 41. 4 � 5a7 � 2ax2 � 3ax5

37. 2x � 3x2 � 1 �1 � 2x � 3x2 38. 9x3 � 7 � 3x5 7 � 9x3 � 3x5

39. c2x3 � c3x2 � 8c 8c � c3x2 � c2x3 40. x3 � 4a � 5a2x6 4a � x3 � 5a2x6

41. 4 � 3ax5 � 2ax2 � 5a7 42. 10x3y2 � 3x9y � 5y4 � 2x2

43. 3xy2 � 4x3 � x2y � 6y 44. �8a5x � 2ax4 � 5 � a2x2

6y � 3xy2 � x2y � 4x3 �5 � 8a5x � a2x2 � 2ax4

Arrange the terms of each polynomial so that the powers of x are in descending order.

45. 5 � x5 � 3x3 x5 � 3x3 � 5 46. 2x � 1 � 6x2 6x2 � 2x � 147. 4a3x2 � 5a � 2a2x3 2a2x3 � 4a3x2 � 5a 48. b2 � x2 � 2xb x2 � 2xb � b2

49. c2 � cx3 � 5c3x2 � 11x 50. 9x2 � 3 � 4ax3 � 2a2x

51. 8x � 9x2y � 7y2 � 2x4 52. 4x3y � 3xy4 � x2y3 � y4

�2x4 � 9x2y � 8x � 7y2 4x 3y � x2y 3 � 3xy 4 � y 4

53. MONEY Write a polynomial to represent the value of q quarters, d dimes, and n nickels. 0.25q � 0.10d � 0.05n

54. MULTIPLE BIRTHS The number of quadruplet births Q in the United Statesfrom 1989 to 1998 can be modeled by Q � �0.5t3 � 11.7t2 � 21.5t � 218.6,where t represents the number of years since 1989. For what values of t does this model no longer give realistic data? Explain your reasoning. t 15; For t 15, the number of quadruplet births declines dramatically.

PACKAGING For Exercises 55 and 56, use the following information.A convenience store sells milkshakes in cups with semisphericallids. The volume of a cylinder is the product of �, the square of the radius r, and the height h. The volume of a sphere is the

product of �43

�, �, and the cube of the radius.

55. Write a polynomial that represents the volume of the container.

56. If the height of the container is 6 inches and the radius is 2 inches, find the volume of the container. 92.15 in3

r

h

r

r

�

x yr

a

x x

x

x x

xx

xb

b

h

Multiple BirthsFrom 1980 to 1997, thenumber of triplet andhigher births rose 404%(from 1377 to 6737 births).This steep climb in multiplebirths coincides with theincreased use of fertilitydrugs.Source: National Center for

Health and Statistics

0.5xy � �r 2

�

�

55. �r 2h � �23

��r 3

42. 5y4 � 2x2 �10x3y2 � 3x9y49. cx 3 � 5c3x2 �11x � c2

50. 4ax3 � 9x2 �2a2x � 3




ELL



Modeling Write severalmonomials of different degreeson large pieces of paper. Askstudent volunteers to stand infront of the class holding themonomials for the class to see.Have the class arrange thevolunteers in ascending and thendescending order, according tothe degree of the monomial thatthey are holding.

Getting Ready for Lesson 8-5PREREQUISITE SKILL Students willlearn how to add and subtractpolynomials in Lesson 8-5. Thiswill include simplifying expres-sions and combining like terms.Use Exercises 72–76 to determineyour students’ familiarity withsimplifying expressions.

Assessment Options


Mid-Chapter Test (Lessons 8-1through 8-4) is available on p. 519 of the Chapter 8 ResourceMasters.





Express each number in scientific notation. (Lesson 8-3)

61. 12,300,000 62. 0.00345 63. 12 � 106 64. 0.77 � 10�10

1.23 107 3.45 10�3 1.2 107 7.7 10�11

Simplify. Assume that no denominator is equal to zero. (Lesson 8-2)

65. a0b�2c�1 �b12c� 66. �

�n5

8n5� �

�n3

5� 67. ��4x

3

3

zy2

��2

�16

9xz

6

2y4

� 68. �(�

y3

ym)5

�

m7

8

� �y2m15

Determine whether each relation is a function. (Lesson 4-6)

69. no 70. yes

71. PROBABILITY A card is selected at random from a standard deck of 52 cards.What is the probability of selecting a black card? (Lesson 2-6) �1

2�

PREREQUISITE SKILL Simplify each expression. If not possible, write simplified.(To review evaluating expressions, see Lesson 1-5.)

72. 3n � 5n 8n 73. 9a2 � 3a � 2a2 7a2 � 3a 74. 12x2 � 8x � 6

75. �3a � 5b � 4a � 7b a � 2b 76. 4x � 3y � 6 � 7x � 8 � 10y11x � 7y � 2

y

xO

57. CRITICAL THINKING Tell whether the following statement is true or false.Explain your reasoning. True; see margin for explanation.The degree of a binomial can never be zero.

58. Answer the question that was posed at the beginning ofthe lesson. See pp. 471A-471B.

How are polynomials useful in modeling data?


• a discussion of the accuracy of the equation by evaluating the polynomial for t � {0, 1, 2, 3, 4, 5}, and

• an example of how and why someone might use this equation.

59. If x � �1, then 3x3 � 2x2 � x � 1 � B�5. �1. 1. 2.

60. QUANTITATIVE COMPARISON Compare the quantity in Column A and thequantity in Column B. Then determine whether:

the quantity in Column A is greater,the quantity in Column B is greater,the two quantities are equal, orthe relationship cannot be determined from the information given. CD

C

B

A

DCBA

WRITING IN MATH

Mixed Review

Column A Column B

the degree of 5x 2y3 the degree of 3x 3y 2

x y

�2 �2

0 1

3 4

5 �2

74. simplified


4 Assess4 Assess

About the Exercises…Organization by Objective• Degree of a Polynomial:

15–36, 53–56• Writing Polynomials in

Order: 37–52


Assignment GuideBasic: 15–53 odd, 57–76

Average: 15–53 odd, 55–76


Answer

57. For the degree of a binomial to be zero, the highestdegree of either term would need to be zero. Thusboth terms must be of degree zero and are thereforelike terms. With these like terms combined, theexpression is not a binomial, but a monomial.Therefore, the degree of a binomial can never bezero. Only a monomial can have a degree of zero.

AlgebraActivity


TeachTeach


Objective Use algebra tiles toadd and subtract polynomials.

Materialsalgebra tiles

• Stress the concept of a zero pair.Have students form zero pairs

using 1 tiles, x tiles, and x2 tiles.The concept of a zero pair isessential in future modelingactivities.

• Activity 1 Tell students that it iseasier to use the tiles to modelthe polynomials if they arrangethe tiles in the same order thatthe monomials are arrangedwithin each polynomial. In thiscase, the monomials arearranged in descending order.Therefore, arrange the tiles indescending order from left toright on the mat. The x2 tilesgo on the left, the x tiles go inthe middle, and the 1 tiles goon the right.

• Show students how the resultof the model can show a wayto add the polynomials.

Algebra Activity Adding and Subtracting Polynomials 437


Monomials such as 5x and �3x are called liketerms because they have the same variable to the same power. When you use algebra tiles, you can recognize like terms because the individual tiles have the same size and shape.

Adding and Subtracting Polynomials

Activity 1 Use algebra tiles to find (3x2 � 2x � 1) � (x2 � 4x � 3).

Model each polynomial.

3x2 � 2x � 1 →

x2 � 4x � 3 →

Combine like terms and remove zero pairs.

Write the polynomial for the tiles that remain.

(3x2 � 2x � 1) � (x2 � 4x � 3) � 4x2 � 2x � 2

x 2 � �4x �3

3x 2 �2x� � 1

�x �x

x 2

x 2 x 2 x 2

1

x x x x�1�1�1

2x� � �24x 2

�x �xx 2 x 2

x 2 x 2 x x x x�1�1�1

1

Polynomial Models

Like terms are represented by tiles that have the same shape and size.

A zero pair may be formed by pairing one tile with its opposite. You can remove or add zero pairs without changing the polynomial.

x �x O

x x �x

like terms

Algebra Activity Adding and Subtracting Polynomials 437

Teaching Algebra withManipulatives• pp. 10–11 (master for algebra tiles)• p. 137 (student recording sheet)

Glencoe Mathematics Classroom Manipulative Kit• algebra tiles


3 x2 � 2 x � 1

(�) 1 x2 � 4 x � 3

4 x2 � 2 x � 2


AssessAssess


Algebra ActivityAlgebra Activity

5x � 4

xx xxx11 1 1

7x � 1

xx xxxx x1

�x �x1 1 1

Model and AnalyzeUse algebra tiles to find each sum or difference. 1–6. See margin.1. (5x2 � 3x � 4) � (2x2 � 4x � 1) 2. (2x2 � 5) � (3x2 � 2x � 6) 3. (�4x2 � x) � (5x � 2)

4. (3x2 � 4x � 2) � (x2 � 5x � 5) 5. (�x2 � 7x) � (2x2 � 3x) 6. (8x � 4) � (6x2 � x � 3)

7. Find (2x2 � 3x � 1) � (2x � 3) using each method from Activity 2 and Activity 3. Illustrate with

drawings and explain in writing how zero pairs are used in each case. See pp. 471A–471B.

Activity 2 Use algebra tiles to find (5x � 4) � (�2x � 3).

Model the polynomial 5x � 4.

To subtract �2x � 3, you

must remove 2 red �x tiles

and 3 yellow 1 tiles. You can

remove the yellow 1 tiles,

but there are no red �x tiles.

Add 2 zero pairs of x tiles.

Then remove the 2 red

�x tiles.


(5x � 4) � (�2x � 3) � 7x � 1

Activity 3 Use algebra tiles and the additive inverse, or opposite, to find (5x � 4) � (�2x � 3).

To find the difference of 5x � 4

and �2x � 3, add 5x � 4 and the

opposite of �2x � 3.

5x � 4 →

The opposite of →�2x � 3 is 2x � 3.


(5x � 4) � (�2x � 3) � 7x � 1 Notice that this is the same answer as in Activity 2.

2x � �3

xx

xxxxx

5x � 4

�1 �1�1

1 1 1 1

Recall that you can subtract a number by adding its additive inverse or opposite.Similarly, you can subtract a polynomial by adding its opposite.


• Activity 2 Explain that addinga zero pair to the polynomialdoes not change its valuebecause the zero pairs areequal to zero. After adding thetwo zero pairs, there areenough red x tiles to remove to satisfy the operation. Theseven remaining green x tilesrepresent 7x.

• Write the difference verticallyso students can see that eachpair of like terms is subtracted.

5x � 4(�) �2x � 3

7x � 1

5 � (�2)↑ ↑ 4 � 3

• Activity 3 Students may findthat it is easier to add theadditive inverse when usingalgebra tiles. By doing so, theycan avoid having to add zeropairs as they had to do inActivity 2.

• Have students reworkExample 3 using the subtrac-tion method in Example 2 toconfirm the result. Then reworkExample 2 using the additiveinverse method.

Ask students to discuss whichmethod seems easier for subtract-ing polynomials. Give students anexercise to find the sum or differ-ence without using models. Havestudents who answer incorrectlyuse models to determine wherethey erred.


Answers

1. 7x2 � x � 3

2. 5x2 � 2x � 11

3. �4x2 � 6x � 2

4. 2x2 � 9x � 7

5. �3x2 � 4x

6. �6x2 � 7x � 7




can adding polynomialshelp you model sales?

Ask students:• Look at the polynomials for

video game and regular toysales. How many terms doeseach polynomial have? 4

• Compare the terms in the twopolynomials. Are the terms liketerms? yes

• How do you think you mightgo about adding the two poly-nomials? Combine the like terms.

• Who would want predictionsof toy sales? Sample answer: Toystores would want to know howmuch more inventory to carry thanthe previous year.

Lesson 8-5 Adding and Subtracting Polynomials 439www.algebra1.com/extra_examples

ADD POLYNOMIALS To add polynomials, you can group like termshorizontally or write them in column form, aligning like terms.

Adding and SubtractingPolynomials

Add Polynomials Find (3x2 � 4x � 8) � (2x � 7x2 � 5).

Method 1 Horizontal

Group like terms together.

(3x2 � 4x � 8) � (2x � 7x2 � 5)

� [3x2 � (�7x2)] � (�4x � 2x) � [8 � (�5)] Associative and Commutative Properties

� �4x2 � 2x � 3 Add like terms.

Method 2 Vertical

Align the like terms in columns and add.

3x2 � 4x � 8 Notice that terms are in descending order

(�) �7x2 � 2x � 5with like terms aligned.

�4x2 � 2x � 3

Example 1Example 1

• Add polynomials.

• Subtract polynomials.

From 1996 to 1999, the amount of sales (in billions of dollars) of video games Vand traditional toys R in the United States can be modeled by the following equations, where t is the number of years since 1996.Source: Toy Industry Fact Book

V � �0.05t3 � 0.05t2 � 1.4t � 3.6R � 0.5t3 � 1.9t2 � 3t � 19

The total toy sales T is the sum of the video game sales V and traditional toy sales R.

Study Tip

SUBTRACT POLYNOMIALS Recall that you can subtract a rational number by adding its opposite or additive inverse. Similarly, you can subtract a polynomial by adding its additive inverse.

To find the additive inverse of a polynomial, replace each term with its additive inverse or opposite.

Polynomial Additive Inverse

�5m � 3n 5m � 3n

2y2 � 6y � 11 �2y2 � 6y � 11

7a � 9b � 4 �7a � 9b � 4

can adding polynomials help you model sales?can adding polynomials help you model sales?

Adding ColumnsWhen adding like terms incolumn form, rememberthat you are addingintegers. Rewrite eachmonomial to eliminatesubtractions. For example, you couldrewrite 3x2 � 4x � 8 as 3x2 � (�4x) � 8.





TechnologyInteractive Chalkboard



Transparencies

LessonNotes

1 Focus1 Focus


33

22


11

In-Class ExampleIn-Class ExampleADD POLYNOMIALS

Find (7y2 � 2y � 3) �

(2 � 4y � 5y2). 12y2 � 2y � 1

Teaching Tip Some studentsmay benefit by marking throughlike terms as they mentallycombine them. This saves timespent rewriting terms so liketerms are grouped.

SUBTRACT POLYNOMIALS

Teaching Tip Explain tostudents that in order tocombine like terms whensubtracting polynomials, youmust add the additive inverse.

Find (6y2 � 8y4 � 5y) �

(9y4 � 7y � 2y2).�y4 � 4y2 � 2y

GEOMETRY The measure ofthe perimeter of the triangleshown is 37s � 42.

a. Find the polynomial thatrepresents the third side ofthe triangle. 13s � 6

b. Find the length of the thirdside of the triangle if s � 3meters. 45 m

Answers

1. The powers of x and y are not thesame.

12. 4n2 � 5

13. 13z � 10z2

14. 2a2 � 6a � 8

15. �2n2 � 7n � 5

16. 5x � 2y � 3

17. 5b3 � 8b2 � 4b

18. 10d2 � 8

14s � 16

10s � 20

When polynomials are used to model real-world data, their sums and differencescan have real-world meaning too.


Subtract Polynomials Find (3n2 � 13n3 � 5n) � (7n � 4n3).

Method 1 Horizontal

Subtract 7n � 4n3 by adding its additive inverse.

(3n2 � 13n3 � 5n) � (7n � 4n3)

� (3n2 � 13n3 � 5n) � (�7n � 4n3) The additive inverse of 7n � 4n3 is �7n � 4n3.

� 3n2 � [13n3 � (�4n3)] � [5n � (�7n)] Group like terms.

� 3n2 � 9n3 � 2n Add like terms.

Method 2 Vertical

Align like terms in columns and subtract by adding the additive inverse.

3n2 � 13n3 � 5n 3n2 � 13n3 � 5n

(�) 4n3 � 7n (�) �4n3 � 7n

3n2 � 9n3 � 2n

Thus, (3n2 � 13n3 � 5n) � (7n � 4n3) � 3n2 � 9n3 � 2n or, arranged in descendingorder, 9n3 � 3n2 � 2n.

Example 2Example 2

Subtract PolynomialsEDUCATION The total number of public school teachers T consists of twogroups, elementary E and secondary S. From 1985 through 1998, the number (in thousands) of secondary teachers and total teachers in the United Statescould be modeled by the following equations, where n is the number of yearssince 1985.

S � 11n � 942T � 44n � 2216

a. Find an equation that models the number of elementary teachers E for thistime period.

You can find a model for E by subtracting the polynomial for S from thepolynomial for T.

Total 44n � 2216 44n � 2216

� Secondary (�) 11n � 942 (�) �11n � 942

Elementary 33n � 1274

An equation is E � 33n � 1274.

b. Use the equation to predict the number of elementary teachers in the year 2010.

The year 2010 is 2010 � 1985 or 25 years after the year 1985.

If this trend continues, the number of elementary teachers in 2010 would be 33(25) � 1274 thousand or about 2,099,000.

Example 3Example 3

Inverse of aPolynomialWhen finding the additive inverse of apolynomial, remember to find the additive inverse of every term.

Study Tip

TeacherThe educationalrequirements for a teaching license vary by state. In 1999, the average public K–12teacher salary was $40,582.

Add the opposite.

Add the opposite.

Online ResearchFor information about a career as a teacher, visit:www.algebra1.com/careers


19. 2g3 � 9g

20. �8y3 � 3y2 � y � 17

21. �2x � 3xy

22. �2x2 � 8x � 8

23. 3ab2 � 11ab � 4

24. x3 � 2x2 � 2x � 6

25. 3x2 � 12x � 5ax � 3a2

PowerPoint®

PowerPoint®

2 Teach2 Teach

http://www.algebra1.com/careers

Lesson 8-5 Adding and Subtracting Polynomials 441

1. Explain why 5xy2 and 3x2y are not like terms. See margin.2. OPEN ENDED Write two polynomials whose difference is 2x2 � x � 3.

3. FIND THE ERROR Esteban and Kendra are finding (5a � 6b) � (2a � 5b).

Who is correct? Explain your reasoning.

Find each sum or difference.

4. (4p2 � 5p) � (�2p2 � p) 2p2 � 6p 5. (5y2 � 3y � 8) � (4y2 � 9)

6. (8cd � 3d � 4c) � (�6 � 2cd) 7. (6a2 � 7a � 9) � (�5a2 � a � 10)

8. (g3 � 2g2 � 5g � 6) � (g2 � 2g) 9. (3ax2 � 5x � 3a) � (6a � 8a2x � 4x)g 3 � 3g2 � 3g � 6 3ax2 � 9x � 9a � 8a2x

POPULATION For Exercises 10 and 11, use the following information.From 1990 through 1999, the female population F and the male population M of theUnited States (in thousands) is modeled by the following equations, where n is thenumber of years since 1990. Source: U.S. Census Bureau

F � 1247n � 126,971 M � 1252n � 120,741

10. Find an equation that models the total population T in thousands of the UnitedStates for this time period. T � 2499n � 247,712

11. If this trend continues, what will the population of the United States be in 2010?about 297,692,000

Kendra

(5a – 6b) – (2a + 5b)

= (5a – 6b) + (–2a – 5b)

= 3a – 1 1b

Esteban

(5a – 6b) – (2a + 5b)

= (–5a + 6b) + (–2a – 5b)

= –7a + b

Concept Check 2. Sample answer: 6x2 � 4x � 7 and 4x2 � 3x � 43. Kendra; Estebanadded the additiveinverses of both polynomials when he should have addedthe opposite of the polynomial being subtracted.

Guided Practice5. 9y2 � 3y � 16. 10cd � 3d � 4c � 67. 11a2 � 6a � 1

Application




Find each sum or difference. 12–25. See margin.12. (6n2 � 4) � (�2n2 � 9) 13. (9z � 3z2) � (4z � 7z2)

14. (3 � a2 � 2a) � (a2 � 8a � 5) 15. (�3n2 � 8 � 2n) � (5n � 13 � n2)

16. (x � 5) � (2y � 4x � 2) 17. (2b3 � 4b � b2) � (�9b2 � 3b3)

18. (11 � 4d2) � (3 � 6d2) 19. (4g3 � 5g) � (2g3 � 4g)

20. (�4y3 � y � 10) � (4y3 � 3y2 � 7) 21. (4x � 5xy � 3y) � (3y � 6x � 8xy)

22. (3x2 � 8x � 4) � (5x2 � 4) 23. (5ab2 � 3ab) � (2ab2 � 4 � 8ab)

24. (x3 � 7x � 4x2 � 2) � (2x2 � 9x � 4) 25. (5x2 � 3a2 � 5x) � (2x2 � 5ax � 7x)

26. (3a � 2b � 7c) � (6b � 4a � 9c) � (�7c � 3a � 2b) �4a � 6b � 5c27. (5x2 � 3) � (x2 � x � 11) � (2x2 � 5x � 7) 8x2 � 6x � 1528. (3y2 � 8) � (5y � 9) � (y2 � 6y � 4) 2y2 � y � 529. (9x3 � 3x � 13) � (6x2 � 5x) � (2x3 � x2 � 8x � 4) 11x3 � 7x2 � 9

GEOMETRY The measures of two sides of a triangle are given. If P is theperimeter, find the measure of the third side.

30. P � 7x � 3y 31. P � 10x2 � 5x � 164x � 2y 6x2 � 15x � 12

10x � 7

4x 2 � 3

�

2x � 3yx � 2y


Exercises Examples12–31 1, 232, 33 3


�

�

�


4–9 1, 210, 11 3



Study NotebookStudy NotebookHave students—• select an exercise and show two

methods for finding the sum ordifference.


FIND THE ERRORHave students

check each step ofEsteban and Kendra’s work.

Remind students that since this isa subtraction problem, Estebanand Kendra need to add theadditive inverse.

About the Exercises…Organization by Objective• Add Polynomials: 12–17,

26, 27, 34–40• Subtract Polynomials:

18–25, 28–33


Assignment GuideBasic: 13–27 odd, 32, 33, 41,44–68

Average: 13–31 odd, 34–38,41–68


Teaching Tip You may wantstudents to make the box describedin Exercises 36–40.

Interpersonal To reinforce the concepts of the lesson, place studentsin pairs and have the students take turns completing the Check forUnderstanding Exercises. As one student works the problem, have theother student offer guidance and suggestions. Make sure students offerconstructive reinforcement to each other and that each studentcompletes at least one exercise.





NAME ______________________________________________ DATE ____________ PERIOD _____

6-58-5

Less

on

8-5

Add Polynomials To add polynomials, you can group like terms horizontally or writethem in column form, aligning like terms vertically. Like terms are monomial terms thatare either identical or differ only in their coefficients, such as 3p and �5p or 2x2y and 8x2y.

Find (2x2 � x � 8) �(3x � 4x2 � 2).

Horizontal MethodGroup like terms.

(2x2 � x � 8) � (3x � 4x2 � 2)� [(2x2 � (�4x2)] � (x � 3x ) � [(�8) � 2)]� �2x2 � 4x � 6.

The sum is �2x2 � 4x � 6.

Find (3x2 � 5xy) �(xy � 2x2).

Vertical MethodAlign like terms in columns and add.

3x2 � 5xy(�) 2x2 � xy Put the terms in descending order.

5x2 � 6xy

The sum is 5x2 � 6xy.


ExercisesExercises

Find each sum.

1. (4a � 5) � (3a � 6) 2. (6x � 9) � (4x2 � 7)

7a � 1 4x2 � 6x � 2

3. (6xy � 2y � 6x) � (4xy � x) 4. (x2 � y2) � (�x2 � y2)

10xy � 5x � 2y 2y2

5. (3p2 � 2p � 3) � (p2 � 7p � 7) 6. (2x2 � 5xy � 4y2) � (�xy � 6x2 � 2y2)

4p2 � 9p � 10 �4x2 � 4xy � 6y2

7. (5p � 2q) � (2p2 � 8q � 1) 8. (4x2 � x � 4) � (5x � 2x2 � 2)

2p2 � 5p � 6q � 1 6x2 � 4x � 6

9. (6x2 � 3x) � (x2 � 4x � 3) 10. (x2 � 2xy � y2) � (x2 � xy � 2y2)

7x2 � x � 3 2x2 � xy � y2

11. (2a � 4b � c) � (�2a � b � 4c) 12. (6xy2 � 4xy) � (2xy � 10xy2 � y2)

�5b � 5c �4xy2 � 6xy � y2

13. (2p � 5q) � (3p � 6q) � (p � q) 14. (2x2 � 6) � (5x2 � 2) � (�x2 � 7)

6p 6x2 � 11

15. (3z2 � 5z) � (z2 � 2z) � (z � 4) 16. (8x2 � 4x � 3y2 � y) � (6x2 � x � 4y)

4z2� 8z � 4 14x2� 3x � 3y2� 5y



1. (4y � 5) � (�7y � 1) 2. (�x2 � 3x) � (5x � 2x2)�3y � 4 �3x2 � 2x

3. (4k2 � 8k � 2) � (2k � 3) 4. (2m2 � 6m) � (m2 � 5m � 7)4k2 � 6k � 1 3m2 � m � 7

5. (2w2 � 3w � 1) � (4w � 7) 6. (g3 � 2g2) � (6g � 4g2 � 2g3)2w2 � w � 6 �g3 � 6g2 � 6g

7. (5a2 � 6a � 2) � (7a2 � 7a � 5) 8. (�4p2 � p � 9) � (p2 � 3p � 1)�2a2 � 13a � 3 �3p2 � 2p � 8

9. (x3 � 3x � 1) � (x3 � 7 � 12x) 10. (6c2 � c � 1) � (�4 � 2c2 � 8c)9x � 6 4c2 � 9c � 5

11. (�b3 � 8bc2 � 5) � (7bc2 � 2 � b3) 12. (5n2 � 3n � 2) � (�n � 2n2 � 4)�2b3 � bc2 � 7 7n2 � 4n � 2

13. (4y2 � 2y � 8) � (7y2 � 4 � y) 14. (w2 � 4w � 1) � (�5 � 5w2 � 3w)�3y2 � 3y � 12 6w2 � 7w � 6

15. (4u2 � 2u � 3) � (3u2 � u � 4) 16. (5b2 � 8 � 2b) � (b � 9b2 � 5)7u2 � 3u � 1 �4b2 � b � 13

17. (4d2 � 2d � 2) � (5d2 � 2 � d) 18. (8x2 � x � 6) � (�x2 � 2x � 3)9d2 � d 9x2 � x � 3

19. (3h2 � 7h � 1) � (4h � 8h2 � 1) 20. (4m2 � 3m � 10) � (m2 � m � 2)�5h2 � 3h � 2 5m2 � 2m � 8

21. (x2 � y2 � 6) � (5x2 � y2 � 5) 22. (7t2 � 2 � t) � (t2 � 7 � 2t)�4x2 � 2y2 � 1 8t2 � 3t � 5

23. (k3 � 2k2 � 4k � 6) � (�4k � k2 � 3) 24. (9j2 � j � jk) � (�3j2 � jk � 4j)k3 � 3k2 � 8k � 9 6j 2 � 3j

25. (2x � 6y � 3z) � (4x � 6z � 8y) � (x � 3y � z) 7x � 5y � 4z

26. (6f 2 � 7f � 3) � (5f 2 � 1 � 2f ) � (2f 2 � 3 � f ) �f 2 � 10f � 1

27. BUSINESS The polynomial s3 � 70s2 � 1500s � 10,800 models the profit a companymakes on selling an item at a price s. A second item sold at the same price brings in aprofit of s3 � 30s2 � 450s � 5000. Write a polynomial that expresses the total profitfrom the sale of both items. 2s3 � 100s2 � 1950s � 15,800

28. GEOMETRY The measures of two sides of a triangle are given.If P is the perimeter, and P � 10x � 5y, find the measure of the third side. 2x � 2y

3x � 4y

5x � y

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____




NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5

Pre-Activity How can adding polynomials help you model sales?


What operation would you use to find how much more the traditional toysales R were than the video games sales V ?

subtraction

Reading the Lesson

1. Use the example (�3x3 � 4x2 � 5x � 1) � (�5x3 � 2x2 � 2x � 7).

a. Show what is meant by grouping like terms horizontally.

[�3x3 � (�5x3)] � [(4x2 � (�2x2)] � (5x � 2x) � [1 � (�7)]

b. Show what is meant by aligning like terms vertically.

�3x3 � 4x2 � 5x � 1(�) �5x3 � 2x2 � 2x � 7

c. Choose one method, then add the polynomials.

�8x3 � 2x2 � 7x � 6

2. How is subtracting a polynomial like subtracting a rational number?

You subtract by adding the additive inverse.

3. An algebra student got the following exercise wrong on his homework. What was his error?

(3x5 � 3x4 � 2x3 � 4x2 � 5) � (2x5 � x3 � 2x2 � 4)� [3x5 � (�2x5)] � (�3x4) � [2x3 � (�x3)] � [�4x2 � (�2x2)] � (5 � 4)� x5 � 3x4 � x3 � 6x2 � 9

He did not add the additive inverse of �x3.


4. How is adding and subtracting polynomials vertically like adding and subtractingdecimals vertically?

Aligning like terms when adding or subtracting polynomials is like usingplace value to align digits when you add or subtract decimals.


Circular Areas and Volumes

Area of Circle Volume of Cylinder Volume of Cone

A � �r2 V � �r2h V � �r2h

Write an algebraic expression for each shaded area. (Recall that the diameter of a circle is twice its radius.)

1. 2. 3.

3x2x2x x

y x x yx

x

r

h

r

h

r

1�3

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____



MoviesIn 1998, attendance at movie theaters was at its highest point in 40 years with 1.48 billiontickets sold for a record$6.95 billion in grossincome. Source: The National Association

of Theatre Owners

�

�

�

�

MOVIES For Exercises 32 and 33, use the following information.From 1990 to 1999, the number of indoor movie screens I and total movie screens Tin the U.S. could be modeled by the following equations, where n is the number ofyears since 1990.

I � 161.6n2 � 20n � 23,326 T � 160.3n2 � 26n � 24,226

32. Find an equation that models the number of outdoor movie screens D in theU.S. for this time period. D � �1.3n2 � 6n � 900

33. If this trend continues, how many outdoor movie screens will there be in theyear 2010? 260 outdoor screens

NUMBER TRICK For Exercises 34 and 35, use the following information.Think of a two-digit number whose ones digit is greater than its tens digit. Multiplythe difference of the two digits by 9 and add the result to your original number.Repeat this process for several other such numbers. 34–35. See margin.

34. What observation can you make about your results?

35. Justify that your observation holds for all such two-digit numbers by letting x equal the tens digit and y equal the ones digit of the original number. (Hint: The original number is then represented by 10x � y.)

POSTAL SERVICE For Exercises 36–40, use the information below and in the figure at the right. The U.S. Postal Service restricts the sizes of boxesshipped by parcel post. The sum of the length and the girth of the box must not exceed 108 inches.

Suppose you want to make an open box using a 60-by-40 inch piece of cardboard bycutting squares out of each corner and folding up the flaps. The lid will be madefrom another piece of cardboard. You do not know how big the squares should be,so for now call the length of the side of each square x.

36. Write a polynomial to represent the length of the box formed. 60 � 2x37. Write a polynomial to represent the width of the box formed. 40 � 2x38. Write a polynomial to represent the girth of the box formed. 80 � 2x39. Write and solve an inequality to find the least possible value of x you could use

in designing this box so it meets postal regulations. 140 � 4x � 108; 8 in.40. What is the greatest integral value of x you could use to design this box if it

does not have to meet regulations? 19 in.

CRITICAL THINKING For Exercises 41–43, suppose x is an integer.

41. Write an expression for the next integer greater than x. x � 1

42. Show that the sum of two consecutive integers, x and the next integer after x, isalways odd. (Hint: A number is considered even if it is divisible by 2.) See margin.

43. What is the least number of consecutive integers that must be added together toalways arrive at an even integer? 4

40 in.

60 in.

fold fold

fold

fold

x

x

x

x

x

x

x

x

length

height

width

girth � 2(width) � 2(height)


ELL


Writing Have students write ashort essay describing how tosubtract monomials using theadditive inverse. Make surestudents include an examplewith their descriptions.

Getting Ready for Lesson 8-6PREREQUISITE SKILL Studentswill learn how to multiply a poly-nomial by a monomial in Lesson8-6. This process is an applica-tion of the Distributive Property.Use Exercises 63–68 to determineyour students’ familiarity withthe Distributive Property.

Answers

42. �

� � or x �

x � (x � 1) when divided by 2 has

a remainder of , therefore the

sum of two consecutive integers is odd.

44. In order to find the sum of thevideo games sales and thetraditional toy sales, you mustadd the two polynomial models Vand R, which represent each ofthese sales from 1996 to 1999.

• T � 0.45t3 � 1.85t2 � 4.4t �22.6

• If a person was looking to investin a toy company, they mightwant to look at the trend in toysales over the last severalyears and try to predict toysales for the future.

55–56.

O

wp

m

605040302010

weeks1 2 3 4 5 6 7 8 9 10

1�2

1�2

1�2

2x�2

2x � 1�

2x � (x � 1)��

2



44. Answer the question that was posed at the beginning ofthe lesson. See margin.

How can adding polynomials help you model sales?


• an equation that models total toy sales, and

• an example of how and why someone might use this equation.

45. The perimeter of the rectangle shown at the right is 16a � 2b. Which of the following expressionsrepresents the length of the rectangle? A

3a � 2b 10a � 2b2a � 3b 6a � 4b

46. If a2 � 2ab � b2 � 36 and a2 � 3ab � b2 � 22, find ab. D6 8 12 14DCBA

DC

BA

5a � b

WRITING IN MATH

Mixed Review


65. 35p � 28q


Find the degree of each polynomial. (Lesson 8-4)

47. 15t3y2 5 48. 24 0 49. m2 � n3 3 50. 4x2y3z � 5x3z 6

Express each number in standard notation. (Lesson 8-3)

51. 8 � 106 52. 2.9 � 105 53. 5 � 10�4 54. 4.8 � 10�7

8,000,000 290,000 0.0005 0.00000048KEYBOARDING For Exercises 55–59, use the table below that shows thekeyboarding speeds and experience of 12 students. (Lesson 5-2)

55–56. See margin.55. Make a scatter plot of these data.

56. Draw a best-fit line for the data.

57. Find the equation of the line. Sample answer: y � 4x � 1758. Use the equation to predict the keyboarding speed of a student after a 12-week

course. about 65 wpm59. Can this equation be used to predict the speed for any number of weeks of

experience? Explain. No; there’s a limit as to how fast one can keyboard.

State the domain and range of each relation. (Lesson 4-3)

60. {(�2, 5), (0, �2), (�6, 3)} 61. {(�4, 2), (�1, �3), (5, 0), (�4, 1)}D � {�2, 0, �6}; R � {5, �2, 3} D � {�4, �1, 5}; R � {2, �3, 0, 1}

62. MODEL TRAINS One of the most popular sizes of model trains is called the

HO. Every dimension of the HO model measures �817� times that of a real engine.

The HO model of a modern diesel locomotive is about 8 inches long. About howmany feet long is the real locomotive? (Lesson 3-6) 58

PREREQUISITE SKILL Simplify. (To review the Distributive Property, see Lesson 1-7.)

63. 6(3x � 8) 18x � 48 64. �2(b � 9) �2b � 18 65. �7(�5p � 4q)

66. 9(3a � 5b � c) 67. 8(x2 � 3x � 4) 68. �3(2a2 � 5a � 7)27a � 45b � 9c 8x2 � 24x � 32 �6a2 � 15a � 21

4 7 8 1 6 3 5 2 9 6 7 10

33 45 46 20 40 30 38 22 52 44 42 55

Experience (weeks)

Keyboarding Speed (wpm)


4 Assess4 Assess

Answers

34. The result is always the original number with its digits swapped.

35. Original number � 10x � y ; show that the new number will always be represented by 10y � x.

new number � 9(y � x) � (10x � y)� 9y � 9x � 10x � y� 10y � x




is finding the productof a monomial and a

polynomial related to finding thearea of a rectangle?Ask students:• What is the formula for finding

the area of a rectangle? The areaof a rectangle is A � � w, where � is the length and w is the width.

• What are � and w for therectangle shown? � is x � 3 andw is 2x.

• What is the area of therectangle? 2x2 � 6x

• Substitute the given values forA, �, and w to write an equationfor the area of this rectangle.2x2 � 6x � (x � 3)(2x)

PRODUCT OF MONOMIAL AND POLYNOMIAL The DistributiveProperty can be used to multiply a polynomial by a monomial.

Multiplying a Polynomial by a Monomial


is finding the product of a monomial and a polynomialrelated to finding the area of a rectangle?is finding the product of a monomial and a polynomialrelated to finding the area of a rectangle?

Multiply a Polynomial by a MonomialFind �2x2(3x2 � 7x � 10).

Method 1 Horizontal

�2x2(3x2 � 7x � 10)

� �2x2(3x2) � (�2x2)(7x) � (�2x2)(10) Distributive Property

� �6x4 � (�14x3) � (�20x2) Multiply.

� �6x4 � 14x3 � 20x2 Simplify.

Method 2 Vertical

3x2 � 7x � 10

(�) �2x2 Distributive Property

�6x4 � 14x3 � 20x2 Multiply.

Example 1Example 1

Simplify ExpressionsSimplify 4(3d2 � 5d) � d(d2 � 7d � 12).

4(3d2 � 5d) � d(d2 � 7d � 12)

� 4(3d2) � 4(5d) � (�d)(d2) � (�d)(7d) � (�d)(12) Distributive Property

� 12d2 � 20d � (�d3) � (�7d2) � (�12d) Product of Powers

� 12d2 � 20d � d3 � 7d2 � 12d Simplify.

� �d3 � (12d2 � 7d2) � (20d � 12d)Commutative and Associative Properties

� �d3 � 19d2 � 8d Combine like terms.

Example 2Example 2

Look BackTo review the DistributiveProperty, see Lesson 1-5.

Study Tip

When expressions contain like terms, simplify by combining the like terms.

← ← ←

The algebra tiles shown are grouped together to forma rectangle with a width of 2x and a length of x � 3.Notice that the rectangle consists of 2 blue x2 tiles and 6 green x tiles. The area of the rectangle is the sum of these algebra tiles or 2x2 � 6x.

x 2

x 2

x x x

x x x

x

x

x

1 1 1

• Find the product of a monomial and a polynomial.

• Solve equations involving polynomials.

LessonNotes

1 Focus1 Focus



School-to-Career Masters, p. 16





Transparencies

Lesson 8-6 Multiplying a Polynomial by a Monomial 445

SOLVE EQUATIONS WITH POLYNOMIAL EXPRESSIONS Manyequations contain polynomials that must be added, subtracted, or multiplied beforethe equation can be solved.


Use Polynomial ModelsPHONE SERVICE Greg pays a fee of $20 a month for local calls. Long-distancerates are 6¢ per minute for in-state calls and 5¢ per minute for out-of-state calls.Suppose Greg makes 300 minutes of long-distance phone calls in January and m of those minutes are for in-state calls.

a. Find an expression for Greg’s phone bill for January.

Words The bill is the sum of the monthly fee, in-state charges, and the out-of-state charges.

Variables If m � number of minutes of in-state calls, then 300 � m � number of minutes of out-of-state calls. Let B � phone bill for the month ofJanuary.

service in-state 6¢ per out-of-state 5¢ perbill = fee � minutes � minute � minutes � minute

Equation B � 20 � m � 0.06 � (300 � m) � 0.05

� 20 � 0.06m � 300(0.05) � m(0.05) Distributive Property

� 20 � 0.06m � 15 � 0.05m Simplify.

� 35 � 0.01m Simplify.

An expression for Greg’s phone bill for January is 35 � 0.01m, where m is thenumber of minutes of in-state calls.

b. Evaluate the expression to find the cost if Greg had 37 minutes of in-state calls in January.

35 � 0.01m � 35 � 0.01(37) m � 37

� 35 � 0.37 Multiply.

� $35.37 Add.

Greg’s bill was $35.37.

� � ��

Example 3Example 3

Example 4Example 4

Phone ServiceAbout 98% of long-distancecompanies service their calls using the network ofone of three companies.Since the quality of phoneservice is basically the same, a company’s rates are the primary factor inchoosing a long-distanceprovider.Source: Chamberland Enterprises

Polynomials on Both SidesSolve y(y � 12) � y(y � 2) � 25 � 2y(y � 5) � 15.

y(y � 12) � y(y � 2) � 25 � 2y(y � 5) � 15 Original equation

y2 � 12y � y2 � 2y � 25 � 2y2 � 10y � 15 Distributive Property

2y2 � 10y � 25 � 2y2 � 10y � 15 Combine like terms.

�10y � 25 � 10y � 15 Subtract 2y2 from each side.

�20y � 25 � �15 Subtract 10y from each side.

�20y � �40 Subtract 25 from each side.

y � 2 Divide each side by �20.

The solution is 2.

CHECK y(y � 12) � y(y � 2) � 25 � 2y(y � 5) �15 Original equation

2(2 � 12) � 2(2 � 2) � 25 � 2(2)(2 � 5) � 15 y � 2

2(�10) � 2(4) � 25 � 4(7) � 15 Simplify.

�20 � 8 � 25 � 28 � 15 Multiply.

13 � 13 � Add and subtract.


2 Teach2 Teach

11

22

33


44


PRODUCT OF MONOMIALAND POLYNOMIAL

Teaching Tip Explain tostudents that the two methodsshown are actually two forms ofthe same method.

Teaching Tip If students arehaving difficulty multiplying by anegative monomial in Example 1,you may want to have studentsapply the negative first (bymultiplying all terms by �1) andthen multiply by 2x2.

Find 6y(4y2 � 9y � 7).24y3 � 54y2 � 42y

Teaching Tip Remind studentsthat they must follow the orderof operations when simplifyingexpressions. In Example 2, youmust multiply before you canadd.

Simplify 3(2t2 � 4t � 15) �6t(5t � 2). 36t2 � 45

ENTERTAINMENT Admissionto the Super Fun AmusementPark is $10. Once in the park,super rides are an additional$3 each and regular rides arean additional $2. Sarita goes tothe park and rides 15 rides, ofwhich s of those 15 are superrides.

a. Find an expression for howmuch money Sarita spent atthe park. 40 � s

b. Evaluate the expression tofind the cost if Sarita rode 9 super rides. $49

SOLVE EQUATIONS WITHPOLYNOMIALEXPRESSIONS

Solve b(12 � b) � 7 � 2b �b(�4 � b) 1

�2

Visual/Spatial Have students analyze the rectangle shown.

Ask: What is the width and length? Write the area as a product. Label each part of the figure by its area. Compare these areas with the product.

baaa

a a 3


PowerPoint®

PowerPoint®



Study NotebookStudy NotebookHave students—• list skills from other lessons that

they use in solving polynomialequations.


Answers

2. The three monomials that makeup the trinomial are similar to thethree digits that make up the 3-digit number. The two monomialsthat make up the binomial aresimilar to the two digits that makeup a 2-digit number. With eachprocedure you are performing atotal of 6 multiplications. Eachmethod is vertical and involveslining up like terms or in the case ofregular numerical multiplication,like place values. The differenceis that polynomial multiplicationinvolves variables and the resultingproduct is often the sum of two ormore monomials while numericalmultiplication results in a singlenumber.

15. 5r2 � r3

16. 2w4 � 9w3

17. �32x � 12x2

18. �10y3 � 35y2

19. 7ag4 � 14a2g2

20. �3n2p � 6np2

21. �6b4 � 8b3 � 18b2

22. 30x3 � 18x4 � 66x5

23. 40x3y � 16x2y3 � 24x2y

24. �3cd3 � 2c3d3 � 4c2d2

25. �15hk4 � h2k2 � 6hk2

26. 4a5b � a3b2 � 6a2b3

27. �10a3b2 � 25a4b2 � 5a3b3 �5a6b

28. 8p4q2 � 4p2q4 � 36p5q2 � 12p2q3

8�3

15�4




1. State the property used in each step to multiply 2x(4x2 � 3x � 5).

2x(4x2 � 3x � 5) � 2x(4x2) � 2x(3x) � 2x(5)

� 8x1 � 2 � 6x1 � 1 � 10x

� 8x3 � 6x2 � 10x Simplify.

2. Compare and contrast the procedure used to multiply a trinomial by a binomialusing the vertical method with the procedure used to multiply a three-digitnumber by a two-digit number. See margin.

3. OPEN ENDED Write a monomial and a trinomial involving a single variable.Then find their product. Sample answer: 4x and x2 � 2x � 3; 4x3 � 8x2 � 12x

Find each product. 5. 18b5 � 27b4 � 9b3 � 72b2

4. �3y(5y � 2) �15y2 � 6y 5. 9b2(2b3 � 3b2 � b � 8)

6. 2x(4a4 � 3ax � 6x2) 7. �4xy(5x2 �12xy � 7y2)

Simplify.

8. t(5t � 9) � 2t 5t2 � 11t 9. 5n(4n3 � 6n2 � 2n � 3) � 4(n2 � 7n)

Solve each equation.

10. �2(w � 1) � w � 7 � 4w 3 11. x(x � 2) � 3x � x(x � 4) � 5 �53

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SAVINGS For Exercises 12–14, use the following information.Kenzie’s grandmother left her $10,000 for college. Kenzie puts some of the moneyinto a savings account earning 4% per year, and with the rest, she buys a certificateof deposit (CD) earning 7% per year.

12. If Kenzie puts x dollars into the savings account, write an expression to representthe amount of the CD. 10,000 � x

13. Write an equation for the total amount of money T Kenzie will have saved forcollege after one year. T � 10,700 � 0.03x

14. If Kenzie puts $3000 in savings, how much money will she have after one year?$10,610

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Distributive Property;Product of PowersProperty

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Concept Check


Exercises Examples15–28 129–38 239–48 449–54, 358–62



4–7 18, 9 2

10, 11 412–14 3

Find each product. 15–28. See margin.15. r(5r � r2) 16. w(2w3 � 9w2) 17. �4x(8 � 3x)

18. 5y(�2y2 � 7y) 19. 7ag(g3 � 2ag) 20. �3np(n2 � 2p)

21. �2b2(3b2 � 4b � 9) 22. 6x3(5 � 3x � 11x2)

23. 8x2y(5x � 2y2 � 3) 24. �cd2(3d � 2c2d � 4c)

25. ��34

�hk2(20k2 � 5h � 8) 26. �23

�a2b(6a3 � 4ab � 9b2)

27. �5a3b(2b � 5ab � b2 � a3) 28. 4p2q2(2p2 � q2 � 9p3 � 3q)

Simplify. 31. 20w2 � 18w � 10 32. 10n4 � 5n3 � n2 � 44n29. d(�2d � 4) � 15d �2d2 � 19d 30. �x(4x2 � 2x) � 5x3 �9x3 � 2x2

31. 3w(6w � 4) � 2(w2 � 3w � 5) 32. 5n(2n3 � n2 � 8) � n(4 � n)

33. 10(4m3 � 3m � 2) � 2m(�3m2 � 7m � 1) 46m3 � 14m2 � 32m � 20 34. 4y(y2 � 8y � 6) � 3(2y3 � 5y2 � 2) �2y3 � 17y2 � 24y � 635. �3c2(2c � 7) � 4c(3c2 � c � 5) � 2(c2 � 4) 6c3 � 23c2 � 20c � 836. 4x2(x � 2) � 3x(5x2 � 2x � 6) � 5(3x2 � 4x) 19x3 � x2 � 2x

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Guided Practice

8a4x � 6ax2 � 12x3 �20x3y � 48x2y2 � 28xy3

20n4 � 30n3 � 14n2 � 13n

Application


55. Let x and y be integers. Then 2x and 2y are even numbers, and (2x)(2y) � 4xy. 4xy is divisible by 2 since one of its factors, 4, is divisible by 2. Therefore 4xy is an even number.

57. Let x and y be integers. Then 2x is an even number and 2y � 1 is an odd number. Their product, 2x(2y � 1), is always even since one of its factors is 2.


GEOMETRY Find the area of each shaded region in simplest form.

37. 38.

6x2 � 8x 15p2 � 8p � 6Solve each equation.

39. 2(4x � 7) � 5(�2x � 9) � 5 �2 40. 2(5a � 12) � �6(2a � 3) � 2 2

41. 4(3p � 9) � 5 � �3(12p � 5) ��13

� 42. 7(8w � 3) � 13 � 2(6w � 7) �12

�

43. d(d � 1) � 4d � d(d � 8) 0 44. c(c � 3) � c(c � 4) � 9c � 16 8

45. y(y � 12) � 8y � 14 � y(y � 4) �74

� 46. k(k � 7) � 10 � 2k � k(k � 6) �23

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47. 2n(n � 4) � 18 � n(n � 5) � n(n � 2) � 7 �5

48. 3g(g � 4) � 2g(g � 7) � g(g � 6) � 28 7

SAVINGS For Exercises 49 and 50, use the following information.Marta has $6000 to invest. She puts x dollars of this money into a savings accountthat earns 3% per year, and with the rest, she buys a certificate of deposit that earns6% per year.

49. Write an equation for the total amount of money T Marta will have after one year. T � �0.03x � 6360

50. Suppose at the end of one year, Marta has a total of $6315. How much money didMarta invest in each account? savings account: $1500; certificate of deposit: $4500

51. GARDENING A gardener plants corn in a garden with a length-to-width ratio of 5:4.Next year, he plans to increase the garden’s area by increasing its length by 12 feet. Write an expression for this new area.20x2 � 48x

52. CLASS TRIP Mr. Smith’s American History class will take taxis from their hotelin Washington, D.C., to the Lincoln Memorial. The fare is $2.75 for the first mileand $1.25 for each additional mile. If the distance is m miles and t taxis areneeded, write an expression for the cost to transport the group. 1.50t � 1.25mt

NUMBER THEORY For Exercises 53 and 54, let x be an odd integer.

53. Write an expression for the next odd integer. x � 254. Find the product of x and the next odd integer. x2 � 2x

CRITICAL THINKING For Exercises 55–57, use the following information.An even number can be represented by 2x, where x is any integer.

55. Show that the product of two even integers is always even. See margin.

56. Write a representation for an odd integer. 2x � 1 or 2x � 1

57. Show that the product of an even and an odd integer is always even. See margin.

5x

4x

5p

6

2p � 1 3p � 4

4x

3x

3x � 2 2x


Class TripInside the Lincoln Memorialis a 19-foot marble statueof the United States’ 16thpresident. The statue isflanked on either side bythe inscriptions of Lincoln’sSecond Inaugural Addressand Gettysburg Address.Source: www.washington.org

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About the Exercises…Organization by Objective• Product of Monomial and

Polynomial: 15–38, 51–54,58–61

• Solve Equations withPolynomial Expressions:39–50, 62


Assignment GuideBasic: 15–35 odd, 39–45 odd,49–51, 55–57, 63–87

Average: 15–47 odd, 51, 53,55–57, 63–87


All: Practice Quiz 2 (1–10)

http://www.washington.org




NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6

Less

on

8-6

Product of Monomial and Polynomial The Distributive Property can be used tomultiply a polynomial by a monomial. You can multiply horizontally or vertically. Sometimesmultiplying results in like terms. The products can be simplified by combining like terms.

Find �3x2(4x2 � 6x � 8).

Horizontal Method�3x2(4x2 � 6x � 8)

� �3x2(4x2) � (�3x2)(6x) � (�3x2)(8)� �12x4 � (�18x3) � (�24x2)� �12x4 � 18x3 � 24x2

Vertical Method4x2 � 6x � 8

(�) �3x2

�12x4 � 18x3 � 24x2

The product is �12x4 � 18x3 � 24x2.

Simplify �2(4x2 � 5x) �x(x2 � 6x).

�2(4x2 � 5x) � x( x2 � 6x)� �2(4x2) � (�2)(5x) � (�x)(x2) � (�x)(6x)� �8x2 � (�10x) � (�x3) � (�6x2)� (�x3) � [�8x2 � (�6x2)] � (�10x)� �x3 � 14x2 � 10x


ExercisesExercises

Find each product.

1. x(5x � x2) 2. x(4x2 � 3x � 2) 3. �2xy(2y � 4x2)

5x2� x3 4x3� 3x2� 2x �4xy2 � 8x3y

4. �2g( g2 � 2g � 2) 5. 3x(x4 � x3� x2) 6. �4x(2x3 � 2x � 3)

�2g3 � 4g2 � 4g 3x5 � 3x4 � 3x3 �8x4� 8x2� 12x

7. �4cx(10 � 3x) 8. 3y(�4x � 6x3� 2y) 9. 2x2y2(3xy � 2y � 5x)

�40cx � 12cx2 �12xy � 18x3y � 6y2 6x3y3 � 4x2y3 � 10x3y2

Simplify.

10. x(3x � 4) � 5x 11. �x(2x2 � 4x) � 6x2

3x2 � 9x �2x3 � 2x2

12. 6a(2a � b) � 2a(�4a � 5b) 13. 4r(2r2 � 3r � 5) � 6r(4r2 � 2r � 8)

4a2 � 4ab 32r 3 � 68r

14. 4n(3n2 � n � 4) � n(3 � n) 15. 2b(b2 � 4b � 8) � 3b(3b2 � 9b � 18)

12n3 � 5n2 � 19n �7b3 � 19b2 � 70b

16. �2z(4z2 � 3z � 1) � z(3z2 � 2z � 1) 17. 2(4x2 � 2x) � 3(�6x2 � 4) � 2x(x � 1)

�11z 3 � 4z2 � z 28x2 � 6x � 12


Find each product.

1. 2h(�7h2 � 4h) 2. 6pq(3p2 � 4q) 3. �2u2n(4u � 2n)�14h3 � 8h2 18p3q � 24pq2 �8u3n � 4u2n2

4. 5jk(3jk � 2k) 5. �3rs(�2s2 � 3r) 6. 4mg2(2mg � 4g)15j 2k2 � 10jk2 6rs3 � 9r2s 8m2g3 � 16mg3

7. � m(8m2 � m � 7) 8. � n2(�9n2 � 3n � 6)

�2m3 � m2 � m 6n4 � 2n3 � 4n2

Simplify.

9. �2�(3� � 4) � 7� 10. 5w(�7w � 3) � 2w(�2w2 � 19w � 2)�6�2 � 15� �4w3 � 3w2 � 19w

11. 6t(2t � 3) � 5(2t2 � 9t � 3) 12. �2(3m3 � 5m � 6) � 3m(2m2 � 3m � 1)2t2 � 63t � 15 9m2 � 7m � 12

13. �3g(7g � 2) � 3(g2 � 2g � 1) � 3g(�5g � 3) �3g2 � 3g � 3

14. 4z2(z � 7) � 5z(z2 � 2z � 2) � 3z(4z � 2) �z3 � 6z2 � 4z


15. 5(2s � 1) � 3 � 3(3s � 2) 8 16. 3(3u � 2) � 5 � 2(2u � 2) �3

17. 4(8n � 3) � 5 � 2(6n � 8) � 1 18. 8(3b � 1) � 4(b � 3) � 9 �

19. h(h � 3) � 2h � h(h � 2) � 12 4 20. w(w � 6) � 4w � �7 � w(w � 9) �7

21. t(t � 4) � 1 � t(t � 2) � 2 22. u(u � 5) � 8u � u(u � 2) � 4 �4

23. NUMBER THEORY Let x be an integer. What is the product of twice the integer addedto three times the next consecutive integer? 5x � 3

INVESTMENTS For Exercises 24–26, use the following information.Kent invested $5,000 in a retirement plan. He allocated x dollars of the money to a bondaccount that earns 4% interest per year and the rest to a traditional account that earns 5%interest per year.

24. Write an expression that represents the amount of money invested in the traditionalaccount. 5,000 � x

25. Write a polynomial model in simplest form for the total amount of money T Kent hasinvested after one year. (Hint: Each account has A � IA dollars, where A is the originalamount in the account and I is its interest rate.) T � 5,250 � 0.01x

26. If Kent put $500 in the bond account, how much money does he have in his retirementplan after one year? $5,245

3�2

1�4

1�2

7�4

1�4

2�3

1�4

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____




NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6

Pre-Activity How is finding the product of a monomial and a polynomial relatedto finding the area of a rectangle?


You may recall that the formula for the area of a rectangle is A � �w. In

this rectangle, � � and w � . How would you

substitute these values in the area formula?

A � (x � 3)(2x)

Reading the Lesson

1. Refer to Lesson 8-6.

a. How is the Distributive Property used to multiply a polynomial by a monomial?

The monomial is multiplied by each term in the polynomial.

b. Use the Distributive Property to complete the following.

2y2(3y2 � 2y � 7) � 2y2( ) � 2y2( ) � 2y2( )

� � �

�3x3(x3 � 2x2 � 3) � � �

� � �

2. What is the difference between simplifying an expression and solving an equation?

Simplifying an expression is combining like terms. Solving an equationis finding the value of the variable that makes the equation true.


3. Use the equation 2x(x � 5) � 3x(x � 3) � 5x(x � 7) � 9 to show how you would explainthe process of solving equations with polynomial expressions to another algebra student.

Use the Distributive Property. 2x2 � 10x � 3x2 � 9x � 5x2 � 35x � 9

Combine like terms. 5x2 � x � 5x2 � 35x � 9

Subtract 5x2 from both sides. �x � 35x � 9

Subtract 35x from both sides. �36x � �9

Divide each side by �36. x � 0.25

9x36x5�3x6

(�3x3)(3)(�3x3)(2x2)�3x3(x3)

14y24y36y4

72y3y2

2xx � 3


Figurate NumbersThe numbers below are called pentagonal numbers. They are the numbers of dots or disksthat can be arranged as pentagons.

1. Find the product n(3n � 1). �

2. Evaluate the product in Exercise 1 for values of n from 1 through 4. 1, 5, 12, 22

3. What do you notice? They are the first four pentagonal numbers.

4 Find the next six pentagonal numbers 35 51 70 92 117 145

n�2

3n2�

21�2

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221251

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____



VOLUNTEERING For Exercises 58 and 59, use the following information.Laura is making baskets of apples and oranges for homeless shelters. She wants toplace a total of 10 pieces of fruit in each basket. Apples cost 25¢ each, and orangescost 20¢ each.

58. If a represents the number of apples Laura uses, write a polynomial model insimplest form for the total amount of money T Laura will spend on the fruit for each basket. T � 2 � 0.05a

59. If Laura uses 4 apples in each basket, find the total cost for fruit. $2.20

SALES For Exercises 60 and 61, use the following information.A store advertises that all sports equipment is 30% off the retail price. In addition,the store asks customers to select and pop a balloon to receive a coupon for anadditional n percent off the already marked down price of one of their purchases.

60. Write an expression for the cost of a pair of inline skates with retail price p afterreceiving both discounts. 0.7p � 0.007np

61. Use this expression to calculate the cost, not including sales tax, of a $200 pair of inline skates for an additional 10 percent off. $126

62. SPORTS You may have noticed that when runners race around a curved track,their starting points are staggered. This is so each contestant runs the samedistance to the finish line.

If the radius of the inside lane is x and each lane is 2.5 feet wide, how far apart should the officials start the runners in the two inside lanes? (Hint: Circumference of a circle: C � 2�r, where r is the radius of the circle)2.5� or about 7.9 ft


How is finding the product of a monomial and a polynomial related tofinding the area of a rectangle?


• the product of 2x and x � 3 derived algebraically, and

• a representation of another product of a monomial and a polynomial usingalgebra tiles and multiplication.

64. Simplify [(3x2 � 2x � 4) � (x2 � 5x � 2)](x � 2). B2x3 � 7x2 � 8x � 4 2x3 � 3x2 � 8x � 12

4x3 � 11x2 � 8x � 4 �4x3 � 11x2 � 8x � 4

65. A plumber charges $70 for the first thirty minutes of each house call plus $4 foreach additional minute that she works. The plumber charges Ke-Min $122 forher time. What amount of time, in minutes, did the plumber work? A

43 48 58 64DCBA

DC

BA

WRITING IN MATH

x � 2.5

x

Finish

Start

VolunteeringApproximately one third of young people in grades7–12 suggested that“working for the good of my community andcountry” and “helpingothers or volunteering”were important futuregoals.Source: Primedia/Roper National

Youth Opinion Survey


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ELL


Modeling Bring to class a phonebill or a newspaper advertisementfor cellular phone service showingsome rate information. Havestudents use the pricing data tocome up with a word problemsimilar to Example 3 on page445. The word problem must besolved by using polynomials.When students have completedtheir problems, have pairs ofstudents check each others’ work.

Getting Ready for Lesson 8-7PREREQUISITE SKILLS Studentswill learn how to multiply poly-nomials in Lesson 8-7. One stepin multiplying polynomials ismultiplying monomials after theDistributive Property is applied.Use Exercises 82–87 to determineyour students’ familiarity withsimplifying expressions contain-ing products of monomials.

Assessment Options

Practice Quiz 2 The quiz pro-vides students with a brief reviewof the concepts and skills inLessons 8-4 through 8-6. Lessonnumbers are given to the right ofthe exercises or instruction linesso students can review conceptsnot yet mastered.


Answers

80.

81. Stem | Leaf1 | 0 4 5 8 8 82 | 0 0 1 1 23 | 0 44 | 3 4 3 |4 � 34

Stem | Leaf3 | 0 2 34 | 5 7 7 8 95 | 1 3 5 6 76 | 2 8 6 |2 � 62



Practice Quiz 2Practice Quiz 2

Find the degree of each polynomial. (Lesson 8-4)

1. 5x4 4 2. �9n3p4 7 3. 7a2 � 2ab2 3 4. �6 � 8x2y2 � 5y3 4

Arrange the terms of each polynomial so that the powers of x are in ascending order. (Lesson 8-4)

5. 4x2 � 9x � 12 � 5x3 6. 2xy4 � x3y5 � 5x5y � 13x2

�12 � 9x � 4x2 � 5x3 2xy4 � 13x2 � x3y5 � 5x5yFind each sum or difference. (Lesson 8-5)

7. (7n2 � 4n � 10) � (3n2 � 8) 8. (3g3 � 5g) � (2g3 � 5g2 � 3g � 1)10n2 � 4n � 2 g3 � 5g2 � 2g � 1

Find each product. (Lesson 8-6)

9. 5a2(3a3b � 2a2b2 � 6ab3) 10. 7x2y(5x2 � 3xy � y) 35x4y � 21x3y2 � 7x2y2

15a5b � 10a4b2 � 30a3b3

Lessons 8-4 through 8-6

Find each sum or difference. (Lesson 8-5) 67. �4y2 � 5y � 366. (4x2 � 5x) � (�7x2 � x) �3x2 � 6x 67. (3y2 � 5y � 6) � (7y2 � 9)

68. (5b � 7ab � 8a) � (5ab � 4a) 69. (6p3 � 3p2 � 7) � (p3 � 6p2 � 2p) 5b � 12ab � 12a 7p3 � 3p2 � 2p � 7

State whether each expression is a polynomial. If the expression is a polynomial,identify it as a monomial, a binomial, or a trinomial. (Lesson 8-4)

70. 4x2 � 10ab � 6 71. 4c � ab � c 72. �7y

� � y2 no 73. �n3

2�

yes; trinomial yes; binomial yes; monomialDefine a variable, write an inequality, and solve each problem. Then check yoursolution. (Lesson 6-3) 74. 6 � 10n � 9n ; {n n � �6}74. Six increased by ten times a number is less than nine times the number.

75. Nine times a number increased by four is no less than seven decreased by thirteen times the number. 9n � 4 � 7 � 13n ; �nn � �

232��

Write an equation of the line that passes through each pair of points. (Lesson 5-4)

76. (�3, �8), (1, 4) 77. (�4, 5), (2, �7) 78. (3, �1), (�3, 2)y � 3x � 1 y � �2x � 3

79. EXPENSES Kristen spent one fifth of her money on gasoline to fill up her car.Then she spent half of what was left for a haircut. She bought lunch for $7.When she got home, she had $13 left. How much money did Kristen haveoriginally? (Lesson 3-4) $50

For Exercises 80 and 81, use each set of data to make a stem-and-leaf plot.(Lesson 2-5)

80. 49 51 55 62 47 32 56 57 48 47 33 68 53 45 30

81. 21 18 34 30 20 15 14 10 22 21 18 43 44 20 1880–81. See margin.

PREREQUISITE SKILL Simplify. (To review products of polynomials, see Lesson 8-1.)

82. (a)(a) a2 83. 2x(3x2) 6x3

84. �3y2(8y2) �24y4 85. 4y(3y) � 4y(6) 12y2 � 24y86. �5n(2n2) � (�5n)(8n) � (�5n)(4) 87. 3p2(6p2) � 3p2(8p) � 3p2(12)

�10n3 � 40n2 � 20n 18p4 � 24p3 � 36p2

Mixed Review

78. y � ��12

�x � �12

�



4 Assess4 Assess

AlgebraActivity


TeachTeach


Activity 1 Use algebra tiles to find (x � 2)(x � 5).

The rectangle will have a width of x � 2 and a length of x � 5. Use algebra tiles to mark off the dimensions on a product mat. Then complete the rectangle with algebra tiles.

The rectangle consists of 1 blue x2 tile, 7 green x tiles, and 10 yellow 1 tiles. The area of the rectangle is x2 � 7x � 10. Therefore, (x � 2)(x � 5) � x2 � 7x � 10.

x � 2

x 2

xx

1

1

1

1

1

1

1

1

1

1

x � 5

x x x x xx

1

1

x 1 1 1 1 1



Multiplying PolynomialsYou can use algebra tiles to find the product of two binomials.

Activity 2 Use algebra tiles to find (x � 1)(x � 4).

The rectangle will have a width of x � 1 and a length of x � 4. Use algebra tiles to mark off the dimensions on a product mat. Then begin to make the rectangle with algebra tiles.

Determine whether to use 4 yellow 1 tiles or 4 red �1 tiles to complete the rectangle. Remember that the numbers at the top and side give the dimensions of the tile needed. The area of each tile is the product of �1 and �1 or 1. This is represented by a yellow 1 tile. Fill in the space with 4 yellow 1 tiles to complete the rectangle.

The rectangle consists of 1 blue x2 tile, 5 red �x tiles, and 4 yellow 1 tiles. The area of the rectangle is x2 � 5x � 4. Therefore, (x � 1)(x � 4) � x2 � 5x � 4.

x � 1x 2

x � 4

�x �x �x �x

�x 1 1 1 1

x � 1x 2

x � 4

�x �x �x �x

�x

x

�1

x �1 �1 �1 �1


Teaching Algebra withManipulatives• pp. 10–11 (master for algebra tiles)• p. 16 (master for equation mat)• p. 144 (student recording sheet)

Glencoe Mathematics Classroom Manipulative Kit• algebra tiles• equation mat


Objective Use algebra tiles tomultiply polynomials.

Materialsalgebra tilesproduct mat

• Some students may benefit fromlaying tiles along the top andside of the product mat tomodel each expression. Thenhave them remove the twofactors before determiningtheir final product.

• Activity 1 Make sure studentsmark the dimensions properlyon the product mat. Since x tiles are rectangular, remindstudents that the long side isthe correct side to use to marka value of x on the mat.

• When students are filling in themats with the tiles, remindthem to look carefully at boththe horizontal and verticaldimensions of each tile on theproduct mat. If both dimensionshave a value of x, then use anx2 tile. If one dimension is x andthe other is 1, then use an x tile.If both dimensions are 1, thenuse a 1 tile.


AssessAssess

• Activity 2 Remind students topay close attention to whetherthe dimensions for each tile arepositive or negative, as thisaffects which tile to use.Determining whether to use apositive or negative tile is justlike determining whether aproduct is positive or negative.If both dimensions are positive,then the tile is positive. If oneis positive and the other isnegative, then the tile isnegative. If both are negativethen the tile is positive.

• Activity 3 As an alternative toremoving zero pairs, havestudents write the expressionbased on the tiles withoutremoving zero pairs. They canthen simplify the expression bycombining like terms.

For Exercise 7, help students tosee that when using the Distribu-tive Property to multiply polyno-mials, each term from the firstpolynomial is multiplied by eachterm from the second polynomial.


Answer

7. By the Distributive Property, (x � 3)(x � 4) �x(x � 4) � 3(x � 4). The top rowrepresents x(x � 4) or x2 � 4x.The bottom row represents 3(x � 4) or 3x � 12.

Algebra Activity Multiplying Polynomials 451

Activity 3 Use algebra tiles to find (x � 3)(2x � 1).

The rectangle will have a width of x � 3 and a length of 2x � 1. Mark off thedimensions on a product mat. Then begin to make the rectangle with algebra tiles.

Determine what color x tiles and whatcolor 1 tiles to use to complete the rectangle. The area of each x tile is the product of x and �1. This is represented by a red �x tile. The area of the 1 tile is represented by the product of 1 and �1 or �1. This is represented by a red �1 tile. Complete the rectangle with 3 red �x tiles and 3 red �1 tiles.

Rearrange the tiles to simplify thepolynomial you have formed. Notice that a zero pair is formed by one positive and one negative x tile.

There are 2 blue x2 tiles, 5 red �x tiles, and 3 red �1 tiles left. In simplest form, (x � 3)(2x � 1) � 2x2 � 5x � 3.

x 2 x 2 x

�1

�1

�1�x �x �x �x �x

�x

x � 3

x 2 x 2

2x � 1

�x�x�x

�x�x�x

x

�1

�1

�1

x � 3

x 2 x 2

2x � 1

�x�x�x

xx

x x 1

�1

�1

�1

Model and AnalyzeUse algebra tiles to find each product. 1. x2 � 5x � 6 2. x2 � 4x � 31. (x � 2)(x � 3) 2. (x � 1)(x � 3) 3. (x � 1)(x � 2) x2 � x � 24. (x � 1)(2x � 1) 5. (x � 2)(2x � 3) 6. (x � 3)(2x � 4)

2x2 � 3x � 1 2x2 � 7x � 6 2x2 � 2x � 127. You can also use the Distributive Property to find

the product of two binomials. The figure at the right shows the model for (x � 3)(x � 4) separated into four parts. Write a sentence or two explaining how this model shows the use of the Distributive Property. See margin.

x 2 x x x x

xxx

1

1

1

1

1

1

1

1

1 1 1 1

Algebra Activity Multiplying Polynomials 451





Students have learned how touse the Distributive Property tomultiply a polynomial by amonomial. Make sure studentsrealize that they are applying thesame process in multiplyingpolynomials.

is multiplyingbinomials similar to

multiplying two-digit numbers?Ask students:• Explain how the Distributive

Property was used in the firststep of this problem. The number6 in 36 was multiplied by 20 and 4.

• Explain how the DistributiveProperty was used in thesecond step of this problem.The number 30 in 36 was multipliedby 20 and 4.

• How is the third step in thismultiplication process similarto combining like terms whenworking with polynomials?The digits in each place value arecombined: first the ones, then thetens, and finally the hundreds.

MULTIPLY BINOMIALS To multiply two binomials, apply the DistributiveProperty twice as you do when multiplying two-digit numbers.

Vocabulary• FOIL method

Multiplying Polynomials


The Distributive PropertyFind (x � 3)(x � 2).

Method 1 Vertical

Example 1Example 1

is multiplying binomials similar to multiplying two-digit numbers?is multiplying binomials similar to multiplying two-digit numbers?

Step 1Multiply by the ones.

24

144

6 � 24 � 6(20 � 4)� 120 � 24 or 144

Step 2Multiply by the tens.

24

144

30 � 24 � 30(20 � 4)� 600 � 120 or 720

Step 3Add like place values.

24

144

864

� 720

� 36

720

� 36� 36

You can multiply two binomials in a similar way.

Look BackTo review the DistributiveProperty, see Lesson 1-7.

Study Tip

Multiply by 2.

x � 3

2x � 6

2(x � 3) � 2x � 6

Multiply by x.

x � 3

2x � 6x2 � 3x

x(x � 3) � x2 � 3x

Add like terms.

x � 3

2x � 6

x2 � 5x � 6

x2 � 3x

(�) x � 2(�) x � 2(�) x � 2

Method 2 Horizontal

(x � 3)(x � 2) � x(x � 2) � 3(x � 2) Distributive Property

� x(x) � x(2) � 3(x) � 3(2) Distributive Property

� x2 � 2x � 3x � 6 Multiply.

� x2 � 5x � 6 Combine like terms.

An alternative method for finding the product of two binomials can be shownusing algebra tiles.

• Multiply two binomials by using the FOIL method.

• Multiply two polynomials by using the Distributive Property.

To compute 24 � 36, we multiply each digit in 24 by each digit in 36, payingclose attention to the place value of each digit.

LessonNotes

1 Focus1 Focus


Graphing Calculator and Spreadsheet Masters, p. 38






Transparencies

FOIL Method for Multiplying Binomials• Words To multiply two binomials, find the sum of the products of

F the First terms,

O the Outer terms,

I the Inner terms, and

L the Last terms.

• ExampleProduct of Product of Product of Product ofFirst terms Outer terms Inner terms Last terms

(x � 3)(x � 2) � (x)(x) � (�2)(x) � (3)(x) � (3)(�2)� x2 � 2x � 3x � 6� x2 � x � 6

←←←←

Lesson 8-7 Multiplying Polynomials 453

Consider the product of x � 3 and x � 2. The rectangle shown below has a lengthof x � 3 and a width of x � 2. Notice that this rectangle can be broken up into foursmaller rectangles.

The product of (x � 2) and (x � 3) is the sum of these four areas.

(x � 3)(x � 2) � (x � x) � (x � �2) � (3 � x) � (3 � �2) Sum of the four areas

� x2 � (�2x) � 3x � (�6) Multiply.

� x2 � x � 6 Combine like terms.

This example illustrates a shortcut of the Distributive Property called the . You can use the FOIL method to multiply two binomials.method

FOIL

x x

x

x · x

�2

x · �2

3 3

x

3 · x

�2

3 · �2

x � 3

x

x �1 �1

1

1

1

x 2

x � 2

�x �x

�1

�1

�1

�1

�1

�1

xxx


FOIL Method Find each product.

a. (x � 5)(x � 7)

F O I L(x � 5)(x � 7) � (x)(x) � (x)(7) � (�5)(x) � (�5)(7) FOIL method

� x2 � 7x � 5x � 35 Multiply.


b. (2y � 3)(6y � 7)

(2y � 3)(6y � 7)

F O I L� (2y)(6y) � (2y)(�7) � (3)(6y) � (3)(�7) FOIL method

� 12y2 � 14y � 18y � 21 Multiply.

� 12y2 � 4y � 21 Combine like terms.

Example 2Example 2

F L

IO

Study Tip

IO

F L

Checking YourWorkYou can check yourproducts in Examples 2aand 2b by reworking eachproblem using theDistributive Property.


2 Teach2 Teach

11

22

In-Class ExamplesIn-Class ExamplesMULTIPLY BINOMIALS

Teaching Tip Students who areless familiar with the DistributiveProperty may wish to use thevertical method for multiplyingbinomials because it is similar tomultiplying two-digit numbers.Suggest that students use themethod with which they aremost comfortable.

Find (y � 8)(y � 4). y2 � 4y � 32

Find each product.

a. (z � 6)(z � 12) z2 � 18z � 72

b. (5x � 4)(2x � 8)10x2 � 32x � 32

Teaching Tip Remind students thatthe FOIL method only works formultiplying two binomials. Tomultiply any other polynomials, youmust use the Distributive Propertydirectly, rather than taking a shortcut.

Auditory/Musical Music can be a powerful memory tool. Suggest thatgroups of students make up a song or rap to explain how to use theFOIL method to multiply binomials. Have the groups perform their songsin front of the class when they are finished.


PowerPoint®


33


44


Teaching Tip Tell studentsthat they must find the sum ofthe bases first before they canmultiply by one-half the height.This is why b1 � b2 is writtenwithin parentheses when theproblem is translated from thewords into an equation.

GEOMETRY The area A of atriangle is one-half the heighth times the base. Write anexpression for the area of thetriangle. 3x2 � 19x � 14 units2

MULTIPLY POLYNOMIALS

Find each product.

a. (3a � 4)(a2 � 12a � 1)3a3 � 32a2 � 45a � 4

b. (2b2 � 7b � 9)(b2 � 3b � 1)2b4 � 13b3 � 28b2 � 20b � 9

Answer

1.

2a. (3x � 4)(2x � 5) � 3x(2x � 5) � 4(2x � 5)� 6x2 � 15x � 8x � 20� 6x2 � 7x � 20

xx 2

x � 3

2x � 1

�1

x

�1�x

x

�1

xx 2 x x

6x � 4

x � 7

MULTIPLY POLYNOMIALS The Distributive Property can be used tomultiply any two polynomials.


FOIL MethodGEOMETRY The area A of a trapezoid is one-half the height h times the sum of the bases, b1 and b2. Writean expression for the area of the trapezoid.

Identify the height and bases.

h � x � 2

b1 � 3x � 7

b2 � 2x � 1

Now write and apply the formula.

Area equals one-half height times sum of bases.

A � �12

� � h � (b1 � b2)

A � �12

�h(b1 � b2) Original formula

� �12

�(x � 2)[(3x � 7) � (2x � 1)] Substitution

� �12

�(x � 2)(5x � 6) Add polynomials in the brackets.

� �12

�[x(5x) � x(�6) � 2(5x) � 2(�6)] FOIL method

� �12

�(5x2 � 6x � 10x � 12) Multiply.

� �12

�(5x2 � 4x � 12) Combine like terms.

� �52

�x2 � 2x � 6 Distributive Property

The area of the trapezoid is �52

�x2 � 2x � 6 square units.

��

3x � 7

x � 2

2x � 1

Example 3Example 3

The Distributive PropertyFind each product.

a. (4x � 9)(2x2 � 5x � 3)

(4x � 9)(2x2 � 5x � 3)

� 4x(2x2 � 5x � 3) � 9(2x2 � 5x � 3) Distributive Property

� 8x3 � 20x2 � 12x � 18x2 � 45x � 27 Distributive Property

� 8x3 � 2x2 � 33x � 27 Combine like terms.

b. (y2 � 2y � 5)(6y2 � 3y � 1)

(y2 � 2y � 5)(6y2 � 3y � 1)

� y2(6y2 � 3y � 1) � 2y(6y2 � 3y � 1) � 5(6y2 � 3y � 1) Distributive Property

� 6y4 � 3y3 � y2 � 12y3 � 6y2 � 2y � 30y2 � 15y � 5 Distributive Property

� 6y4 � 15y3 � 37y2 � 17y � 5 Combine like terms.

Example 4Example 4

CommonMisconceptionA common mistake whenmultiplying polynomialshorizontally is to combineterms that are not alike.For this reason, you mayprefer to multiplypolynomials in columnform, aligning like terms.

Study Tip


PowerPoint®

PowerPoint®

2b. (3x � 4)(2x � 5)� 3x(2x) � 3x(�5) � 4(2x) � 4(�5)� 6x2 � 15x � 8x � 20� 6x2 � 7x � 20

2c. 3x � 4( ) 2x � 5

�15x � 206x2 � 8x

6x2 � 7x � 20

2d.

x 2

�1 �1 �1 �1�1 �1 �1 �1�1 �1 �1 �1�1 �1 �1 �1�1 �1 �1 �1

x 2

x 2

x 2

x 2

�x�x�x�x�x�x�x

x�x�x�x�x�x�x�x�x

xxxxxxx

x 2

xx 2

3x � 4

2x � 5

x x x

�1 �1 �1 �1�x�1 �1 �1 �1�x�1 �1 �1 �1�x�1 �1 �1 �1�x�1 �1 �1 �1�x

xx 2

x 2

�x�x�x�x�x

x 2

x 2

�x�x�x�x�x

x 2 x x x


1. Draw a diagram to show how you would use algebra tiles to find the product of 2x � 1 and x � 3. See margin.

2. Show how to find (3x � 4)(2x � 5) using each method. 2a–d. See margin.a. Distributive Property b. FOIL method

c. vertical or column method d. algebra tiles

3. OPEN ENDED State which method of multiplying binomials you prefer and why. See students’ work.

Find each product. 4–11. See margin.4. (y � 4)(y � 3) 5. (x � 2)(x � 6) 6. (a � 8)(a � 5)

7. (4h � 5)(h � 7) 8. (9p � 1)(3p � 2) 9. (2g � 7)(5g � 8)

10. (3b � 2c)(6b � 5c) 11. (3k � 5)(2k2 � 4k � 3)

12. GEOMETRY The area A of a triangle is half the product of the base b times the height h. Write apolynomial expression that represents the area of the triangle at the right.

or 3x 2 � �72

�x � �32

�6x 2 � 7x � 3��

22x � 3

3x � 1

Concept Check

Guided Practice

Application



4–11 1, 2, 412 3



Find each product. 31–38. See margin.13. (b � 8)(b � 2) 14. (n � 6)(n � 7) 15. (x � 4)(x � 9)

16. (a � 3)(a � 5) 17. (y � 4)(y � 8) 18. (p � 2)(p � 10)

19. (2w � 5)(w � 7) 20. (k � 12)(3k � 2) 21. (8d � 3)(5d � 2)

22. (4g � 3)(9g � 6) 23. (7x � 4)(5x � 1) 24. (6a � 5)(3a � 8)

25. (2n � 3)(2n � 3) 26. (5m � 6)(5m � 6) 27. (10r � 4)(10r � 4)

28. (7t � 5)(7t � 5) 29. (8x � 2y)(5x � 4y) 30. (11a � 6b)(2a � 3b)

31. (p � 4)(p2 � 2p � 7) 32. (a � 3)(a2 � 8a � 5)

33. (2x � 5)(3x2 � 4x � 1) 34. (3k � 4)(7k2 � 2k � 9)

35. (n2 � 3n � 2)(n2 � 5n � 4) 36. (y2 � 7y � 1)(y2 � 6y � 5)

37. (4a2 � 3a � 7)(2a2 � a � 8) 38. (6x2 � 5x � 2)(3x2 � 2x � 4)

GEOMETRY Write an expression to represent the area of each figure.

39. 40.

41. 42.

3x � 4

�

5x � 8

2x � 1

x � 7

4x � 3

3x � 2

x � 4

2x � 5


Exercises Examples13–38 1, 2, 439–42 3


�125�x2 � 3x � 24 units2 (9x2 � 24x � 16)� units2

2x2 � 3x � 20 units2 6x2 � �127�x � 3 units2

13. b2 � 10b � 1614. n2 � 13n � 4215. x2 � 13x � 3616. a2 � 8a � 1517. y2 � 4y � 3218. p2 � 8p � 2019. 2w2 � 9w � 3520. 3k2 � 34k � 2421. 40d2 � 31d � 622. 36g2 � 51g � 1823. 35x2 � 27x � 424. 18a2 � 63a � 4025. 4n2 � 12n � 926. 25m2 � 60m � 3627. 100r2 � 1628. 49t2 � 2529. 40x2 � 22xy � 8y2

30. 22a2 � 21ab � 18b2






About the Exercises…Organization by Objective• Multiply Binomials: 13–30,

39–42, 48–53• Multiplying Polynomials:

31–38, 43–47


Assignment GuideBasic: 13–41 odd, 45–47, 54–77

Average: 13–43 odd, 45–47,49–51, 53–77

Advanced: 14–44 even, 49–52,54–71 (optional: 72–77)

Answers

31. p3 � 6p2 � p � 28

32. a3 � 11a2 � 29a � 15

33. 6x3 � 23x2 � 22x � 5

34. 21k3 � 34k2 � 19k � 36

35. n4 � 2n3 � 17n2 � 22n � 8

36. y4 � y3 � 38y2 � 41y � 5

37. 8a4 � 2a3 � 15a2 � 31a � 56

38. 18x4 � 3x3 � 20x2 � 16x � 8

Answers4. y2 � 7y � 12

5. x2 � 4x � 12

6. a2 � 3a � 40

7. 4h2 � 33h � 35

8. 27p2 � 21p � 2

9. 10g2 � 19g � 56

10. 18b2 � 3bc � 10c2

11. 6k3 � 2k2 � 29k � 15




NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7

Less

on

8-7

Multiply Binomials To multiply two binomials, you can apply the Distributive Propertytwice. A useful way to keep track of terms in the product is to use the FOIL method asillustrated in Example 2.

Find (x � 3)(x � 4).

Horizontal Method(x � 3)(x � 4)

� x(x � 4) � 3(x � 4)� (x)(x) � x(�4) � 3(x)� 3(�4)� x2 � 4x � 3x � 12� x2 � x � 12

Vertical Method

x � 3(�) x � 4

�4x � 12x2 � 3xx2 � x � 12

The product is x2 � x � 12.

Find (x � 2)(x � 5) usingthe FOIL method.

(x � 2)(x � 5)First Outer Inner Last

� (x)(x) � (x)(5) � (�2)(x) � (�2)(5)� x2 � 5x � (�2x) � 10� x2 � 3x � 10

The product is x2 � 3x � 10.


ExercisesExercises

Find each product.

1. (x � 2)(x � 3) 2. (x � 4)(x � 1) 3. (x � 6)(x � 2)

x2 � 5x � 6 x2 � 3x � 4 x2 � 8x � 12

4. (p � 4)(p � 2) 5. (y � 5)(y � 2) 6. (2x � 1)(x � 5)

p2 � 2p � 8 y2 � 7y � 10 2x2 � 9x � 5

7. (3n � 4)(3n � 4) 8. (8m � 2)(8m � 2) 9. (k � 4)(5k � 1)

9n2 � 24n � 16 64m2 � 4 5k2 � 19k � 4

10. (3x � 1)(4x � 3) 11. (x � 8)(�3x � 1) 12. (5t � 4)(2t � 6)

12x2 � 13x � 3 �3x2 � 25x � 8 10t2 � 22t � 24

13. (5m � 3n)(4m � 2n) 14. (a � 3b)(2a � 5b) 15. (8x � 5)(8x � 5)

20m2 � 22mn � 6n2 2a2 � 11ab � 15b2 64x2 � 25

16. (2n � 4)(2n � 5) 17. (4m � 3)(5m � 5) 18. (7g � 4)(7g � 4)

4n2 � 2n � 20 20m2 � 35m � 15 49g2 � 16


Find each product.

1. (q � 6)(q � 5) 2. (x � 7)(x � 4) 3. (s � 5)(s � 6)q2 � 11q � 30 x2 � 11x � 28 s2 � s � 30

4. (n � 4)(n � 6) 5. (a � 5)(a � 8) 6. (w � 6)(w � 9)n2 � 10n � 24 a2 � 13a � 40 w2 � 15w � 54

7. (4c � 6)(c � 4) 8. (2x � 9)(2x � 4) 9. (4d � 5)(2d � 3)4c2 � 10c � 24 4x2 � 10x � 36 8d 2 � 22d � 15

10. (4b � 3)(3b � 4) 11. (4m � 2)(4m � 3) 12. (5c � 5)(7c � 9)12b2 � 7b � 12 16m2 � 4m � 6 35c2 � 10c � 45

13. (6a � 3)(7a � 4) 14. (6h � 3)(4h � 2) 15. (2x � 2)(5x � 4)42a2 � 45a � 12 24h2 � 24h � 6 10x2 � 18x � 8

16. (3a � b)(2a � b) 17. (4g � 3h)(2g � 3h) 18. (4x � y)(4x � y)6a2 � 5ab � b2 8g2 � 18gh � 9h2 16x2 � 8xy � y2

19. (m � 5)(m2 � 4m � 8) 20. (t � 3)(t2 � 4t � 7)m3 � 9m2 � 12m � 40 t 3 � 7t2 � 19t � 21

21. (2h � 3)(2h2 � 3h � 4) 22. (3d � 3)(2d2 � 5d � 2)4h3 � 12h2 � 17h � 12 6d 3 � 21d 2 � 9d � 6

23. (3q � 2)(9q2 � 12q � 4) 24. (3r � 2)(9r2 � 6r � 4)27q3 � 18q2 � 12q � 8 27r 3 � 36r 2 � 24r � 8

25. (3c2 � 2c � 1)(2c2 � c � 9) 26. (2�2 � � � 3)(4�2 � 2� � 2)6c4 � 7c3 � 27c2 � 17c � 9 8�4 � 8�3 � 10�2 � 4� � 6

27. (2x2 � 2x � 3)(2x2 � 4x � 3) 28. (3y2 � 2y � 2)(3y2 � 4y � 5)4x4 � 12x3 � 8x2 � 6x � 9 9y4 � 6y3 � 17y2 � 18y � 10

GEOMETRY Write an expression to represent the area of each figure.

29. 4x2 � 2x � 2 units2 30. 4x2 � 3x � 1 units2

31. NUMBER THEORY Let x be an even integer. What is the product of the next twoconsecutive even integers? x2 � 6x � 8

32. GEOMETRY The volume of a rectangular pyramid is one third the product of the areaof its base and its height. Find an expression for the volume of a rectangular pyramidwhose base has an area of 3x2 � 12x � 9 square feet and whose height is x � 3 feet.x3 � 7x2 � 15x � 9 feet3

3x � 2

5x � 4

x � 1

4x � 2

2x � 2

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____




NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7

Pre-Activity How is multiplying binomials similar to multiplying two-digitnumbers?


In your own words, explain how the distributive property is used twice tomultiply two-digit numbers.

The ones of the first factor are multiplied by the tens and theones of the other factor. Then the tens of the first factor aremultiplied by the tens and ones of the other factor.

Reading the Lesson

1. How is multiplying binomials similar to multiplying two-digit numbers?

Binomials have two terms and each term of one binomial is multiplied byeach term of the other binomial.

2. Complete the table using the FOIL method.

Product of�

Product of�

Product of�

Product of First Terms Outer Terms Inner Terms Last Terms

(x � 5)(x � 3) � (x)(x) � (x)(�3) � (5)(x) � (5)(�3)

� x2 � �3x � 5x � �15

� x2 � 2x � 15

(3y � 6)(y � 2) � (3y)(y) � (3y)(�2) � (6)(y) � (6)(�2)

� 3y2 � �6y � 6y � �12

� 3y2 � 12


3. Think of a method for remembering all the product combinations used in the FOILmethod for multiplying two binomials. Describe your method using words or a diagram.Sample answer: Imagine that the two binomials are written on the floor.For FOIL, think of all the possible ways you could have your left foot ona term of the first binomial and your right foot on a term of the secondbinomial. Your feet could be on the first terms, the outer terms, the innerterms, or the last terms.


Pascal’s TriangleThis arrangement of numbers is called Pascal’s Triangle.It was first published in 1665, but was known hundreds of years earlier.

1. Each number in the triangle is found by adding two numbers.What two numbers were added to get the 6 in the 5th row?

3 and 3

2. Describe how to create the 6th row of Pascal’s Triangle.

The first and last numbers are 1. Evaluate 1 � 4, 4 � 6, 6 � 4, and 4 � 1to find the other numbers.

3. Write the numbers for rows 6 through 10 of the triangle.

Row 6: 1 5 10 10 5 1Row 7: 1 6 15 20 15 6 1Row 8: 1 7 21 35 35 21 7 1

11 1

1 2 11 3 3 1

1 4 6 4 1

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____



GEOMETRY The volume V of a prism equals the area of the base B times theheight h. Write an expression to represent the volume of each prism.

43. 44.

2a3 � 10a2 � 2a � 10 units3

NUMBER THEORY For Exercises 45–47, consider three consecutive integers. Let the least of these integers be a. 45. a3 � 3a2 � 2a45. Write a polynomial representing the product of these three integers.

46. Choose an integer for a. Find their product. Sample answer: a � 1; 1(2)(3) � 647. Evaluate the polynomial in Exercise 45 for the value of a you chose in

Exercise 46. Describe the result.Sample answer: 6; The result is the same as the product in Exercise 46.

48. BASKETBALL The dimensions of a professional basketball court arerepresented by a width of 2y � 10 feet and a length of 5y � 6 feet. Find an expression for the area of the court. 10y2 � 38y � 60 ft2

OFFICE SPACE For Exercises 49–51, use the following information.Latanya’s modular office is square. Her office in the company’s new building will be 2 feet shorter in one direction and 4 feet longer in the other.

49. Write expressions for the dimensions of Latanya’s new office. x � 2, x � 450. Write a polynomial expression for the area of her new office. x2 � 2x � 851. Suppose her office is presently 9 feet by 9 feet. Will her new office be bigger or

smaller than her old office and by how much? bigger; 10 ft2

52. MENTAL MATH One way to mentally multiply 25 and 18 is to find (20 � 5)(20 � 2). Show how the FOIL method can be used to find each product.

a. 35(19) b. 67(102) c. 8�12

� � 6�34

� d. 12�35

� � 10�23

�

a–d. See margin.53. POOL CONSTRUCTION A homeowner is installing

a swimming pool in his backyard. He wants its length to be 4 feet longer than its width. Then he wants to surround it with a concrete walkway 3 feet wide. If he can only afford 300 square feet of concrete for the walkway, what should the dimensions of the pool be?20 ft by 24 ft

54. CRITICAL THINKING Determine whether the following statement is sometimes, always, or nevertrue. Explain your reasoning. Sometimes; see margin for explanation.The product of a binomial and a trinomial is a polynomial with four terms.

w � 4

w

3 ft

3 ft

3 ft

3 ft

5y � 6 ft

2y � 10 ft

7y � 3

3y

3y 2y

6

�

2a � 2

a � 5

a � 1

BasketballMore than 200 millionpeople a year pay to seebasketball games. That ismore admissions than forany other American sport.Source: Compton’s Encyclopedia

�

�

�

63y3 � 57y2

� 36y units3


ELL


Speaking Have studentvolunteers explain orally howthe FOIL method for multiplyingbinomials works, and use it tosolve a problem.

Getting Ready for Lesson 8-8PREREQUISITE SKILL Studentswill learn squares and productsof sums and differences inLesson 8-8. They need to be ableto find squares of monomialsmentally to efficiently computethese special products. UseExercises 72–77 to determineyour students’ familiarity withpowers of powers and powers ofproducts.

Answers

54. The product of x � 1 and x2 � 2x � 3 is x3 � 3x2 � 5x � 3,which has 4 terms; the product ofy � 1 and x3 � 2x2 � 3x is x3y � 2x2y �3xy � x3 � 2x2 � 3x,which has 6 terms.

55. Multiplying binomials and two-digit numbers each involve the useof the Distributive Property twice.Each procedure involves four multi-plications and the addition of liketerms. Answers should includethe following.

• 24 � 36� (4 � 20)(6 � 30)� (4 � 20)6 � (4 � 20)30� (24 � 120) � (120 � 600)� 144 � 720� 864

• The like terms in vertical two-digit multiplication are digitswith the same place value.



55. Answer the question that was posed at the beginning of the lesson. See margin.

How is multiplying binomials similar to multiplying two-digit numbers?


• a demonstration of a horizontal method for multiplying 24 � 36, and

• an explanation of the meaning of “like terms” in the context of vertical two-digit multiplication.

56. (x � 2)(x � 4) � (x � 4)(x � 2) � C0 2x2 � 4x � 16 �4x 4x

57. The expression (x � y)(x2 � xy � y2) is equivalent to which of the following? Bx2 � y2 x3 � y3 x3 � xy2 x3 � x2y � y2DCBA

DCBA

WRITING IN MATH

Mixed Review




58. 3d(4d2 � 8d � 15) 59. �4y(7y2 � 4y � 3) 60. 2m2(5m2 � 7m � 8)12d3 � 24d2 � 45d �28y3 � 16y2 � 12y 10m4 � 14m3 � 16m2


61. 3x(2x � 4) � 6(5x2 � 2x � 7) 62. 4a(5a2 � 2a � 7) � 3(2a2 � 6a � 9)36x2 � 42 20a3 � 2a2 � 10a � 27

GEOMETRY For Exercises 63 and 64, use the following information. The sum of the degree measures of the angles of a triangle is 180. (Lesson 8-5)

63. Write an expression to represent the measure of the third angle of the triangle. (181 � 7x)°

64. If x � 15, find the measures of the three angles of the triangle. 31°, 73°, 76°

65. Use the graph at the right to determine whether the system below has no solution, one solution, or infinitely many solutions. If the system has onesolution, name it. (Lesson 7-1)

x � 2y � 0y � 3 � �x one; (�6, 3)

If f(x) � 2x � 5 and g(x) � x2 � 3x, find each value. (Lesson 4-6)

66. f(�4) �13 67. g(�2) � 7 5 68. f(a � 3) 2a � 1

Solve each equation or formula for the variable specified. (Lesson 3-8)

69. a � �vt� for t t � �

av

� 70. ax � by � 2cz for y 71. 4x � 3y � 7 for y

y � �ax �

b2cz

� y � ��43

�x � �73

�

PREREQUISITE SKILL Simplify.(To review Power of a Power and Power of a Product Properties, see Lesson 8-1.)

72. (6a)2 36a2 73. (7x)2 49x2 74. (9b)2 81b2

75. (4y2)2 16y 4 76. (2v3)2 4v6 77. (3g4)2 9g8

y

xO

y � 3 � �x

x � 2y � 0

(2x � 1)˚ (5x � 2)˚


4 Assess4 Assess

52a. Sample answer:(30 � 5)(10 � 9) � (30)(10) � 30(9) � 5(10) � 5(9)

� 300 � 270 � 50 � 45 � 665

52b. Sample answer:(60 � 7)(100 � 2) � 60(100) � 60(2) � 7(100) � 7(2)

� 6000 � 120 � 700 � 14 � 6834

52c. Sample answer:

�8 � ��6 � � � 8(6) � 8� � � (6) � � �� 48 � 6 � 3 � � 57

52d. Sample answer:

�12 � ��10 � � � 12(10) � 12� � � (10) � � �� 120 � 8 � 6 � � 134 2

�5

2�5

2�3

3�5

3�5

2�3

2�3

3�5

3�8

3�8

3�4

1�2

1�2

3�4

3�4

1�2




is the product of twobinomials also a

binomial?Ask students:• Looking at the second example,

what sometimes happens whenusing the FOIL method tomultiply binomials thatproduces a binomial product?The like x terms combine toproduce a zero pair, which causesthe term to drop out of the product.

WhenWhen

SQUARES OF SUMS AND DIFFERENCES While you can always use the FOIL method to find the product of two binomials, some pairs of binomials have products that follow a specific pattern. One such pattern is the square of a sum, (a � b)2 or (a � b)(a � b). You can use the diagram below to derive the pattern for this special product.

Vocabulary• difference of squares

Special Products

• Find squares of sums and differences.

• Find the product of a sum and a difference.

Square of a Sum


In the previous lesson, you learned how to multiply two binomials using theFOIL method. You may have noticed that the Outer and Inner terms oftencombine to produce a trinomial product.

F O I L

(x � 5)(x � 3) � x2 � 3x � 5x � 15


This is not always the case, however. Examine the product below.

F O I L

(x � 3)(x � 3) � x2 � 3x � 3x � 9


� x2 � 9 Simplify.

Notice that the product of x � 3 and x � 3 is a binomial.

• Words The square of a � b is the square of a plus twice the product of a andb plus the square of b.

• Symbols (a � b)2 � (a � b)(a � b)� a2 � 2ab � b2

• Example (x � 7)2 � x2 � 2(x)(7) � 72

� x2 � 14x � 49

a � bab

ab

ab

aba2

a 2

b2

b2

a � b

a

a

b

b

� � � �

(a � b)2 � a2 � ab � ab � b2

� a2 � 2ab � b2

WhenWhen is the product of two binomials also a binomial?is the product of two binomials also a binomial?

LessonNotes

1 Focus1 Focus







Transparencies

Square of a Difference• Words The square of a � b is the square of a minus twice the product of a

and b plus the square of b.

• Symbols (a � b)2 � (a � b)(a � b)� a2 � 2ab � b2

• Example (x � 4)2 � x2 � 2(x)(4) � 42

� x2 � 8x � 16

To find the pattern for the square of a difference, (a � b)2, write a � b as a � (�b)and square it using the square of a sum pattern.

(a � b)2 � [a � (�b)]2

� a2 � 2(a)(�b) � (�b)2 Square of a Sum

� a2 � 2ab � b2 Simplify. Note that (�b)2 � (�b)(�b) or b2.

The square of a difference can be found by using the following pattern.

Lesson 8-8 Special Products 459www.algebra1.com/extra_examples

Square of a SumFind each product.

a. (4y � 5)2

(a � b)2 � a2 � 2ab � b2 Square of a Sum

(4y � 5)2 � (4y)2 � 2(4y)(5) � 52 a � 4y and b � 5

� 16y2 � 40y � 25 Simplify.

CHECK Check your work by using the FOIL method.

(4y � 5)2 � (4y � 5)(4y � 5)

F O I L� (4y)(4y) � (4y)(5) � 5(4y) � 5(5)

� 16y2 � 20y � 20y � 25

� 16y2 � 40y � 25 �

b. (8c � 3d)2


(8c � 3d)2 � (8c)2 � 2(8c)(3d) � (3d)2 a � 8c and b � 3d

� 64c2 � 48cd � 9d2 Simplify.

Example 1Example 1

Square of a DifferenceFind each product.

a. (6p � 1)2

(a � b)2 � a2 � 2ab � b2 Square of a Difference

(6p � 1)2 � (6p)2 � 2(6p)(1) � 12 a � 6p and b � 1

� 36p2 � 12p � 1 Simplify.

b. (5m3 � 2n)2


(5m3 � 2n)2 � (5m3)2 � 2(5m3)(2n) � (2n)2 a � 5m3 and b � 2n

� 25m6 � 20m3n � 4n2 Simplify.

Example 2Example 2

(a � b)2In the pattern for (a � b)2,a and b can be numbers,variables, or expressionswith numbers andvariables.

Study Tip

Lesson 8-8 Special Products 459

2 Teach2 Teach

11

22


SQUARES OF SUMS ANDDIFFERENCES

Teaching Tip Remind studentsthat when a or b are expressionswith numbers and variables, theymust square both the constantand variable parts of the term.

Find each product.

a. (7z � 2)2 49z2 � 28z � 4

b. (5q � 9r)2 25q2 � 90qr � 81r2

Find each product.

a. (3c � 4)2 9c2 � 24c � 16

b. (6e � 6f )2 36e2 � 72ef � 36f 2

InterventionSince the squareof a sum and asquare of thedifference are

the same except for the sign ofthe middle term, the risk ofmaking a careless mistakewhen finding the sum of asquare or difference is high.Tell students that they mustpay close attention to the signswhen finding squares of sumsor differences.

New

PowerPoint®


33


44


GEOMETRY Write anexpression that represents thearea of a square that has aside length of 2x � 12 units.4x2 � 48x � 144 units2

PRODUCT OF A SUM AND A DIFFERENCE

Find each product.

a. (9d � 4)(9d � 4) 81d2 � 16

b. (10g � 13h3)(10g � 13h3)

100g2 � 169h6

Teaching Tip Even though it isimportant to learn the specialproducts, remind students thatthey can always find theseproducts using methods fromprevious lessons in this chapter.

Product of a Sum and a Difference

Ed: page isapprox. 5p3

short

• Words The product of a � b and a � b is the square of a minus the square of b.

• Symbols (a � b)(a � b) � (a � b)(a � b)

� a2 � b2

• Example (x � 9)(x � 9) � x2 � 92

� x2 � 81

PRODUCT OF A SUM AND A DIFFERENCE You can use the diagrambelow to find the pattern for the product of a sum and a difference of the same twoterms, (a � b)(a � b). Recall that a � b can be rewritten as a � (�b).

�

�

� a2 � (�b2)

� a2 � b2

The resulting product, a2 � b2, has a special name. It is called a . Notice that this product has no middle term.squares

difference of


Apply the Sum of a SquareGENETICS The Punnett square shows the possible gene combinations of a cross between two pea plants. Each plant passes along one dominant gene T for tallness and one recessive gene t for shortness.

Show how combinations can be modeled by the square of a binomial. Then determine what percent of the offspring will be pure tall, hybrid tall, and pure short.

Each parent has half the genes necessary for tallness and half the genes necessary for shortness. The makeup of each parent can be modeled by 0.5T � 0.5t. Their offspring can be modeled by the product of 0.5T � 0.5tand 0.5T � 0.5t or (0.5T � 0.5t)2.

If we expand this product, we can determine the possible heights of the offspring.


(0.5T � 0.5t)2 � (0.5T)2 � 2(0.5T)(0.5t) � (0.5t)2 a � 0.5T and b � 0.5t

� 0.25T2 � 0.5Tt � 0.25t2 Simplify.

� 0.25TT � 0.5Tt � 0.25tt T2 � TT and t2 � tt

Thus, 25% of the offspring are TT or pure tall, 50% are Tt or hybrid tall, and 25%are tt or pure short.

Example 3Example 3

More About . . .

Geneticist Laboratory geneticists workin medicine to find curesfor disease, in agricultureto breed new crops andlivestock, and in policework to identify criminals.

TT

Tt

Tt

T

T

t

t

pure tall

Tthybrid tall

Tthybrid tall

t tpure short

a � b

a � (�b)

a

a

b

b

ab

a2

�b2

�ababa 2

�b2�ab� � �

zero pair

�a 2�b2

Online ResearchFor information about a career as a geneticist, visit:www.algebra1.com/careers


Verbal/Linguistic Have students copy the key concepts of how to findthe square of a sum, the square of a difference, and the product of asum and a difference onto note cards. Then, when they are workingexercises involving these types of problems, they can flip through theirnote cards for a quick reminder on how to solve the problems.

Differentiated Instruction ELL

PowerPoint®

PowerPoint®

http://www.algebra1.com/careers

Special Products


The following list summarizes the special products you have studied.

Guided Practice

Application

Product of a Sum and a DifferenceFind each product.

a. (3n � 2)(3n � 2)

(a � b)(a � b) � a2 � b2 Product of a Sum and a Difference

(3n � 2)(3n � 2) � (3n)2 � 22 a � 3n and b � 2

� 9n2 � 4 Simplify.

b. (11v � 8w2)(11v � 8w2)


(11v � 8w2)(11v � 8w2) � (11v)2 � (8w2)2 a � 11v and b � 8w2

� 121v2 � 64w4 Simplify.

Example 4Example 4

• Square of a Sum (a � b)2 � a2 � 2ab � b2

• Square of a Difference (a � b)2 � a2 � 2ab � b2

• Product of a Sum and a Difference (a � b)(a � b) � a2 � b2


5–10 1, 2, 411, 12 3

1. Compare and contrast the pattern for the square of a sum with the pattern forthe square of a difference.

2. Explain how the square of a difference and the difference of squares differ.

3. Draw a diagram to show how you would use algebra tiles to model the productof x � 3 and x � 3, or (x � 3)2.

4. OPEN ENDED Write two binomials whose product is a difference of squares.Sample answer: x � 1 and x � 1

Find each product.

5. (a � 6)2 a2 � 12a � 36 6. (4n � 3)(4n � 3) 16n2 � 24n � 97. (8x � 5)(8x � 5) 64x2 � 25 8. (3a � 7b)(3a � 7b) 9a2 � 49b2

9. (x2 � 6y)2 x4 � 12x2y � 36y2 10. (9 � p)2 81 � 18p � p2

GENETICS For Exercises 11 and 12, use the following information.In hamsters, golden coloring G is dominant over cinnamon coloring g. Suppose a purebred cinnamon male is mated with a purebred golden female.

11. Write an expression for the genetic makeup of the hamster pups. 1.0Gg

12. What is the probability that the pups will have cinnamon coloring? Explain your reasoning.0%; All pups will be golden since only Ggcombinations are possible.

Concept Check 1–3. See margin.

Golden

Cinnamon



Study NotebookStudy NotebookHave students—• complete the definitions/examples

for the remaining terms on theirVocabulary Builder worksheets forChapter 8.


About the Exercises…Organization by Objective• Squares of Sums and

Differences: 13–16, 19–22,25–28, 31–32, 35–36, 39–46

• Product of a Sum and aDifference: 17–18, 23–24,29–30, 33–34, 37–38, 47


Assignment GuideBasic: 13–35 odd, 39–44,48–50, 52–70

Average: 13–37 odd, 39–44,47–50, 52–70 (optional: 51)

Advanced: 14–38 even, 45–70

Answers

1. The patterns are the same except fortheir middle terms. The middle termshave different signs.

2. The square of a difference is (a � b)2,which equals a2 � 2ab � b2. Thedifference of squares is the product of a � b and a � b or a2 � b2.

3.

�xx 2

x � 3

x � 31

�x

1

�x

1�x1 1 1�x1 1 1�x


Special Products

NAME ______________________________________________ DATE ____________ PERIOD _____

8-88-8

Less

on

8-8

Squares of Sums and Differences Some pairs of binomials have products thatfollow specific patterns. One such pattern is called the square of a sum. Another is called thesquare of a difference.

Square of a sum (a � b)2 � (a � b)(a � b) � a2 � 2ab � b2

Square of a difference (a � b)2 � (a � b)(a � b) � a2 � 2ab � b2

Find (3a � 4)(3a � 4).

Use the square of a sum pattern, with a �3a and b � 4.

(3a � 4)(3a � 4) � (3a)2 � 2(3a)(4) � (4)2

� 9a2 � 24a � 16

The product is 9a2 � 24a � 16.

Find (2z � 9)(2z � 9).

Use the square of a difference pattern witha � 2z and b � 9.

(2z � 9)(2z � 9) � (2z)2 � 2(2z)(9) � (9)(9)� 4z2 � 36z � 81

The product is 4z2 � 36z � 81.


ExercisesExercises

Find each product.

1. (x � 6)2 2. (3p � 4)2 3. (4x � 5)2

x2 � 12x � 36 9p2 � 24p � 16 16x2 � 40x � 25

4. (2x � 1)2 5. (2h � 3)2 6. (m � 5)2

4x2 � 4x � 1 4h2 � 12h � 9 m2 � 10m � 25

7. (c � 3)2 8. (3 � p)2 9. (x � 5y)2

c2 � 6c � 9 9 � 6p � p2 x2 � 10xy � 25y2

10. (8y � 4)2 11. (8 � x)2 12. (3a � 2b)2

64y2 � 64y � 16 64 � 16x � x2 9a2 � 12ab � 4b2

13. (2x � 8)2 14. (x2 � 1)2 15. (m2 � 2)2

4x2 � 32x � 64 x4 � 2x2 � 1 m4 � 4m2 � 4

16. (x3 � 1)2 17. (2h2 � k2)2 18. � x � 3�2

x6 � 2x3� 1 4h4 � 4h2k2 � k4 x2 � x � 9

19. (x � 4y2)2 20. (2p � 4q)2 21. � x � 2�2

x2 � 8xy2 � 16y4 4p2 � 16pq � 16q2 x2 � x � 48�3

4�9

2�3

3�2

1�16

1�4


Find each product.

1. (n � 9)2 2. (q � 8)2 3. (� � 10)2

n2 � 18n � 81 q2 � 16q � 64 �2 � 20� � 100

4. (r � 11)2 5. ( p � 7)2 6. (b � 6)(b � 6)r 2 � 22r � 121 p2 � 14p � 49 b2 � 36

7. (z � 13)(z � 13) 8. (4e � 2)2 9. (5w � 4)2

z2 � 169 16e2 � 16e � 4 25w2 � 40w � 16

10. (6h � 1)2 11. (3s � 4)2 12. (7v � 2)2

36h2 � 12h � 1 9s2 � 24s � 16 49v2 � 28v � 4

13. (7k � 3)(7k � 3) 14. (4d � 7)(4d � 7) 15. (3g � 9h)(3g � 9h)49k2 � 9 16d 2 � 49 9g2 � 81h2

16. (4q � 5t)(4q � 5t) 17. (a � 6u)2 18. (5r � s)2

16q2 � 25t 2 a2 � 12au � 36u2 25r2 � 10rs � s2

19. (6c � m)2 20. (k � 6y)2 21. (u � 7p)2

36c2 � 12cm � m2 k2 � 12ky � 36y2 u2 � 14up � 49p2

22. (4b � 7v)2 23. (6n � 4p)2 24. (5q � 6s)2

16b2 � 56bv � 49v2 36n2 � 48np � 16p2 25q2 � 60qs � 36s2

25. (6a � 7b)(6a � 7b) 26. (8h � 3d)(8h � 3d) 27. (9x � 2y2)2

36a2 � 49b2 64h2 � 9d 2 81x2 � 36xy2 � 4y4

28. (3p3 � 2m)2 29. (5a2 � 2b)2 30. (4m3 � 2t)2

9p6 � 12p3m � 4m2 25a4 � 20a2b � 4b2 16m6 � 16m3t � 4t 2

31. (6e3 � c)2 32. (2b2 � g)(2b2 � g) 33. (2v2 � 3e2)(2v2 � 3e2)36e6 � 12e3c � c2 4b4 � g2 4v4 � 12v2e2 � 9e4

34. GEOMETRY Janelle wants to enlarge a square graph that she has made so that a sideof the new graph will be 1 inch more than twice the original side s. What trinomialrepresents the area of the enlarged graph? 4s2 � 4s � 1

GENETICS For Exercises 35 and 36, use the following information.In a guinea pig, pure black hair coloring B is dominant over pure white coloring b. Supposetwo hybrid Bb guinea pigs, with black hair coloring, are bred.

35. Find an expression for the genetic make-up of the guinea pig offspring.0.25BB � 0.50Bb � 0.25bb

36. What is the probability that two hybrid guinea pigs with black hair coloring will producea guinea pig with white hair coloring? 25%

Practice (Average)

Special Products

NAME ______________________________________________ DATE ____________ PERIOD _____



Special Products

NAME ______________________________________________ DATE ____________ PERIOD _____

8-88-8

Pre-Activity When is the product of two binomials also a binomial?

Read the introduction to Lesson 8-8 at the top of page 458 in your textook.

What is a meant by the term trinomial product?

a three-term polynomial answer when multiplying polynomials

Reading the Lesson

1. Refer to the Key Concepts boxes on pages 458, 459, and 460.

a. When multiplying two binomials, there are three special products. What are the threespecial products that may result when multiplying two binomials?

square of a sum, square of a difference, product of a sum and adifference

b. Explain what is meant by the name of each special product.

square of a sum: squaring the sum of two monomials; square of adifference: squaring the difference of two monomials; product of asum and a difference: the product of the sum and the difference of thesame two terms

c. Use the examples in the Key Concepts boxes to complete the table.

Symbols Product Example Product

Square of a Sum (a � b)2 a2 � 2ab � b2 (x � 7)2 x2 � 14x � 49

Square of a Difference

(a � b)2 a2 � 2ab � b2 (x � 4)2 x2 � 8x � 16

Product of a Sum and a Difference

(a � b)(a � b) a2 � b2 (x � 9)(x � 9) x2 � 81

2. What is another phrase that describes the product of the sum and difference of two terms? difference of squares


3. Explain how FOIL can help you remember how many terms are in the special productsstudied in this lesson. For the square of a sum and the square of adifference, the inner and outer products are equal, so there are are onlythree terms. For the product of the sum and difference of two terms, twoof the products for FOIL are opposites. That means that the final producthas only two terms.


Sums and Differences of Cubes Recall the formulas for finding some special products:

Perfect-square trinomials: (a � b)2 � a2 � 2ab � b2 or (a � b)2 � a2 � 2ab � b2

Difference of two squares: (a � b)(a � b) � a2 � b2

A pattern also exists for finding the cube of a sum (a � b)3.

1. Find the product of (a � b)(a � b)(a � b).

a3 � 3a2b � 3ab2 � b3

2. Use the pattern from Exercise 1 to evaluate (x � 2)3.

x3 � 6x2 � 12x � 8

3. Based on your answer to Exercise 1, predict the pattern for the cube of a difference (a � b)3.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____



Architecture The historical Gwennap Pit,an outdoor amphitheaterin southern England,consists of a circular stagesurrounded by circularlevels used for seating.Each seating level is about1 meter wide.Source: Christian Guide to Britain



Find each product. 25–38. See margin.13. (y � 4)2 y2 � 8y � 16 14. (k � 8)(k � 8) 15. (a � 5)(a � 5)

16. (n � 12)2 17. (b � 7)(b � 7) b2 � 49 18. (c � 2)(c � 2) c2 � 419. (2g � 5)2 20. (9x � 3)2 21. (7 � 4y)2

22. (4 � 6h)2 23. (11r � 8)(11r � 8) 24. (12p� 3)(12p � 3)

25. (a � 5b)2 26. (m � 7n)2 27. (2x � 9y)2

28. (3n � 10p)2 29. (5w � 14)(5w � 14) 30. (4d � 13)(4d � 13)

31. (x3 � 4y)2 32. (3a2 � b2)2 33. (8a2 � 9b3)(8a2 � 9b3)

34. (5x4 � y)(5x4 � y) 35. ��23

�x � 6�2

36. ��45

�x � 10�2

37. (2n � 1)(2n � 1)(n � 5) � 38. (p � 3)(p � 4)(p � 3)(p � 4)

GENETICS For Exercises 39 and 40, use the following information.Pam has brown eyes and Bob has blue eyes. Brown genes B are dominant over bluegenes b. A person with genes BB or Bb has brown eyes. Someone with genes bb hasblue eyes. Suppose Pam’s genes for eye color are Bb. 39. 0.5Bb � 0.5b2

39. Write an expression for the possible eye coloring of Pam and Bob’s children.

40. What is the probability that a child of Pam and Bob would have blue eyes? �12

�

MAGIC TRICK For Exercises 41–44, use the following information.Julie says that she can perform a magic trick with numbers. She asks you to pick awhole number, any whole number. Square that number. Then, add twice youroriginal number. Next add 1. Take the square root of the result. Finally, subtract youroriginal number. Then Julie exclaims with authority, “Your answer is 1!”

41. Pick a whole number and follow Julie’s directions. Is your result 1?

42. Let a represent the whole number you chose. Then, find a polynomialrepresentation for the first three steps of Julie’s directions. a2 � 2a � 1

43. The polynomial you wrote in Exercise 42 is the square of what binomial sum?

44. Take the square root of the perfect square you wrote in Exercise 43, then subtracta, your original number. What is the result? 1

ARCHITECTURE For Exercises 45 and 46, use the following information.A diagram of a portion of the Gwennap Pit is shown at the right. Suppose the radius of the stage is s meters. 45. s � 2, s � 3

45. Use the information at the left to find binomial representations for the radii of the second and third seating levels.

46. Find the area of the shaded region representing the third seating level. (6.3s � 15.7) m2

47. GEOMETRY The area of the shaded region models the difference of two squares, a2 � b2. Show that the area of the shaded region is also equal to (a � b)(a � b). (Hint: Divide the shaded region into two trapezoids as shown.)See pp. 471A–471B.

a

a

b

b

s


Exercises Examples13–38 1, 2, 439, 40 3


�

�

�

�

14. k2 � 16k � 6415. a2 � 10a � 2516. n2 � 24n � 14419. 4g2 � 20g � 2520. 81x2 � 54x � 921. 49 � 56y � 16y2

22. 16 � 48h � 36h2

23. 121r2 � 6424. 144p2 � 9

41. Sample answer: 2; yes43. (a � 1)2


ELL


Writing Have students write ashort explanation of whyknowing the special productsthat they learned in this lessoncould be important.

Assessment Options


Answers

25. a2 � 10ab � 25b2

26. m2 � 14mn � 49n2

27. 4x2 � 36xy � 81y2

28. 9n2 � 60np � 100p2

29. 25w2 � 19630. 16d2 � 16931. x6 � 8x3y � 16y2

32. 9a4 � 6a2b2 � b4

33. 64a4 � 81b6

34. 25x8 � y2

35. x2 � 8x � 36

36. x2 � 16x � 100

37. 4n3 � 20n2 � n � 538. p4 � 25p2 � 14448. The product of two binomials is

also a binomial when the twobinomials are the sum and thedifference of the same two terms.Answers should include thefollowing.• Sample answer:

(2x � 13)(2x � 13)

• Sample answer: (10x � 11)(10x � 11)

51c. (a � b)3

ab

a b

a

b

16�25

4�9


Extending the Lesson




48. Answer the question that was posed at the beginning of the lesson. See margin.

When is the product of two binomials also a binomial?


• an example of two binomials whose product is a binomial, and

• an example of two binomials whose product is not a binomial.

49. If a2 � b2 � 40 and ab � 12, find the value of (a � b)2. C1 121 16 28

50. If x � y � 10 and x � y � 20, find the value of x2 � y2. B400 200 100 30

51. Does a pattern exist for the cube of a sum, (a � b)3? a. a3 � 3a2b � 3ab2 � b3

a. Investigate this question by finding the product of (a � b)(a � b)(a � b).

b. Use the pattern you discovered in part a to find (x � 2)3. x3 � 6x2 � 12x � 8c. Draw a diagram of a geometric model for the cube of a sum. See margin.

DCBA

DCBA

WRITING IN MATH


52. (x � 2)(x � 7) 53. (c � 9)(c � 3) 54. (4y � 1)(5y � 6)

55. (3n � 5)(8n � 5) 56. (x � 2)(3x2 � 5x � 4) 57. (2k � 5)(2k2 � 8k � 7)24n2 � 25n � 25 3x3 � 11x2 � 14x � 8 4k3 � 6k2 � 26k � 35

Solve. (Lesson 8-6)

58. 6(x � 2) � 4 � 5(3x � 4) 4 59. �3(3a � 8) � 2a � 4(2a � 1) �43

�

60. p(p � 2) � 3p � p(p � 3) 0 61. y(y � 4) � 2y � y(y � 12) � 7 �12

�

Use elimination to solve each system of equations. (Lessons 7-3 and 7-4)

62. �34

�x � �15

�y � 5 (0, 25) 63. 2x � y � 10 (3, �4) 64. 2x � 4 � 3y (5, �2)

�34

�x � �15

�y � �55x � 3y � 3 3y � x � �11

Write the slope-intercept form of an equation that passes through the given pointand is perpendicular to the graph of each equation. (Lesson 5-6)

65. 5x � 5y � 35, (�3, 2) 66. 2x � 5y � 3, (�2, 7) 67. 5x � y � 2, (0, 6)y � x � 5 y � �2.5x � 2 y � �

15

�x � 6Find the nth term of each arithmetic sequence described. (Lesson 4-7)

68. a1 � 3, d � 4, n � 18 71 69. �5, 1, 7, 13, … for n � 12 61

70. PHYSICAL FITNESS Mitchell likes to exercise regularly. He likes to warm up bywalking two miles. Then he runs five miles. Finally, he cools down by walkingfor another mile. Identify the graph that best represents Mitchell’s heart rate as a function of time. (Lesson 1-8) ba. b. c.

Heart

Rate

Time

Heart

Rate

Time

Mixed Review 52. x2 � 9x � 1453. c2 � 6c � 2754. 20y2 � 29y � 6

Heart

Rate

Time


4 Assess4 Assess


Study Guide and Review


Choose a term from the vocabulary list that best matches each example.

1. 4�3 � �413� negative exponent 2. (n3)5 � n15 Power of a Power

3. �4

8

xx

2

yy3� � �

2xy2� Quotient of Powers 4. 4x2 monomial

5. x2 � 3x � 1 trinomial 6. 20 � 1 zero exponent7. x4 � 3x3 � 2x2 � 1 polynomial 8. (x � 3)(x � 4) � x2 � 4x � 3x � 12 FOIL method9. x2 � 2 binomial 10. (a3b)(2ab2) � 2a4b3 Product of Powers

See pages410–415.

Multiplying MonomialsConcept Summary Examples

• A monomial is a number, a variable, or a product 6x2, �5, �23c�

of a number and one or more variables.

• To multiply two powers that have the same base, add exponents. a2 � a3 � a5

• To find the power of a power, multiply exponents. (a2)3 � a6

• The power of a product is the product of the powers. (ab2)3 � a3b6

1 Simplify (2ab2)(3a2b3).

(2ab2)(3a2b3) � (2 � 3)(a � a2)(b2 � b3) Commutative Property

� 6a3b5 Product of Powers

2 Simplify (2x2y3)3.

(2x2y3)3 � 23(x2)3(y3)3 Power of a Product

� 8x6y9 Power of a Power

Exercises Simplify. See Examples 2, 3, and 5 on pages 411 and 412.

11. y3 � y3 � y y7 12. (3ab)(�4a2b3) �12a3b4 13. (�4a2x)(�5a3x4) 20a5x5

14. (4a2b)3 64a6b3 15. (�3xy)2(4x)3 576x5y2 16. (�2c2d)4(�3c2)3 �432c14d4

17. ��12

�(m2n4)2 ��12

�m4n8 18. (5a2)3 � 7(a6) 132a6 19. [(32)2]3 531,441

8-18-1

Vocabulary and Concept CheckVocabulary and Concept Check

www.algebra1.com/vocabulary_review

binomial (p. 432)constant (p. 410)degree of a monomial (p. 433)degree of a polynomial (p. 433)difference of squares (p. 460)FOIL method (p. 453)

monomial (p. 410)negative exponent (p. 419)polynomial (p. 432)Power of a Power (p. 411)Power of a Product (p. 412)Power of a Quotient (p. 418)

Product of Powers (p. 411)Quotient of Powers (p. 417)scientific notation (p. 425)trinomial (p. 432)zero exponent (p. 419)

ExamplesExamples


Have students look through the chapter to make sure they haveincluded examples in their Foldables for each type of polynomialand monomial operation they learned.Encourage students to refer to their Foldables while completingthe Study Guide and Review and to use them in preparing for theChapter Test.

TM

For more informationabout Foldables, seeTeaching Mathematicswith Foldables.

Lesson-by-LessonReviewLesson-by-LessonReview

Vocabulary and Concept CheckVocabulary and Concept Check

• This alphabetical list ofvocabulary terms in Chapter 8includes a page referencewhere each term wasintroduced.

• Assessment A vocabularytest/review for Chapter 8 isavailable on p. 516 of theChapter 8 Resource Masters.

For each lesson,

• the main ideas aresummarized,

• additional examples reviewconcepts, and

• practice exercises are provided.

The Vocabulary PuzzleMakersoftware improves students’ mathematicsvocabulary using four puzzle formats—crossword, scramble, word search using aword list, and word search using clues.Students can work on a computer screenor from a printed handout.

Vocabulary PuzzleMaker

ELL

MindJogger Videoquizzesprovide an alternative review of conceptspresented in this chapter. Students workin teams in a game show format to gainpoints for correct answers. The questionsare presented in three rounds.

Round 1 Concepts (5 questions)Round 2 Skills (4 questions)Round 3 Problem Solving (4 questions)

MindJogger Videoquizzes

ELL



Chapter 8 Study Guide and Review 465

Dividing MonomialsConcept Summary Examples

• To divide two powers that have the same base, subtract �aa

5

3� � a2

the exponents.

• To find the power of a quotient, find the power of the ��ba

��2

� �ba2

2�

numerator and the power of the denominator.

• Any nonzero number raised to the zero power is 1. (3a3b2)0 � 1

• For any nonzero number a and any integer n, a�3 � �a13�

a�n � �a1n� and �

a�1

n� � an.

Simplify �82xx2

6

yy

2�. Assume that x and y are not equal to zero.

�8

2

xx2

6

yy2� � ��

28

��xx

6

2��yy2�� Group the powers with the same base.

� ��14

��(x6 � 2)(y1 � 2) Quotient of Powers

� �4xy

4� Simplify.

Exercises Simplify. Assume that no denominator is equal to zero.See Examples 1–4 on pages 417–420.

20. �(3

6

ya)0

� �61a� 21. ��34bdc

2��

3�2674bd

3c3

6� 22. x�2y0z3 �

xz3

2�

23. �2174bb

�

�

2

3� �2174b

� 24. �(138aa

3

2bbc3

2

c)4

2� �

2ab

4� 25. �

�

�

1

4

6

8

aa

3

4

bb

2

xxy

4

3

y� �

3baxy

3

2�

26. �(�

aa5b)5

2b8

� �b6 27. �(4

(a2

�

a

1

4))

�

2

2� �

641a6� 28. ��35

5

xx�

y2

�

y

2

�6��01

Chapter 8 Study Guide and ReviewChapter 8 Study Guide and Review

ExampleExample

See pages417–423.

8-28-2

See pages425–430.

8-38-3 Scientific Notation Concept Summary

• A number is expressed in scientific notation when it is written as a product of a factor and a power of 10. The factor must be greater than or equal to 1 and less than 10.

a � 10n, where 1 � a � 10 and n is an integer.

1 Express 5.2 107 in standard notation.

5.2 � 107 � 52,000,000 n � 7; move decimal point 7 places to the right.

2 Express 0.0021 in scientific notation.

0.0021 → 0002.1 � 10n Move decimal point 3 places to the right.

0.0021 � 2.1 � 10�3 a � 2.1 and n � �3

ExamplesExamples




See pages432–436.

8-48-4

Chapter 8 Study Guide and ReviewChapter 8 Study Guide and Review

Polynomials Concept Summary

• A polynomial is a monomial or a sum of monomials.

• A binomial is the sum of two monomials, and a trinomial is the sum ofthree monomials.

• The degree of a monomial is the sum of the exponents of all its variables.

• The degree of the polynomial is the greatest degree of any term. To findthe degree of a polynomial, you must find the degree of each term.

1 Find the degree of 2xy3 � x2y.

Polynomial Terms Degree of Each Term Degree of Polynomial2xy3 � x2y 2xy3, x2y 4, 3 4

2 Arrange the terms of 4x2 � 9x3 � 2 � x so that the powers of x are indescending order.

4x2 � 9x3 � 2 � x � 4x2 � 9x3 � 2x0 � x1 x0 � 1 and x � x1

� 9x3 � 4x2 � x � 2 3 2 1 0

Exercises Find the degree of each polynomial. See Example 3 on page 433.

38. n � 2p2 2 39. 29n2 � 17n2t2 4 40. 4xy � 9x3z2 � 17rs3 541. �6x5y � 2y4 � 4 � 8y2 42. 3ab3 � 5a2b2 � 4ab 4 43. 19m3n4 � 21m5n 7

6Arrange the terms of each polynomial so that the powers of x are in descendingorder. See Example 5 on page 433. 45. �4x4 � 5x3y2 � 2x2y3 � xy � 2744. 3x4 � x � x2 � 5 3x4 � x2 � x � 545. �2x2y3 � 27 � 4x4 � xy � 5x3y2

3 Evaluate (2 102)(5.2 106). Express the result in scientific and standard notation.

(2 � 102)(5.2 � 106) � (2 � 5.2)(102 � 106) Associative Property

� 10.4 � 108 Product of Powers

� (1.04 � 101) � 108 10.4 = 1.04 � 101

� 1.04 � (101 � 108) Associative Property

� 1.04 � 109 or 1,040,000,000 Product of Powers

Exercises Express each number in standard notation. See Example 1 on page 426.

29. 2.4 � 105 240,000 30. 3.14 � 10�4 0.000314 31. 4.88 � 109 4,880,000,000

Express each number in scientific notation. See Example 2 on page 426.

32. 0.00000187 33. 796 � 103 34. 0.0343 � 10�2

1.87 10�6 7.96 105 3.43 10�4

Evaluate. Express each result in scientific and standard notation.See Examples 3 and 4 on page 427.

35. (2 � 105)(3 � 106) 36. �18..44 �

�1100

�

�

6

9� 37. (3 � 102)(5.6 � 10�8)6 1011; 6 103; 6000 1.68 10�5;600,000,000,000 0.0000168

ExamplesExamples


Study Guide and ReviewChapter 8 Study Guide and ReviewChapter 8 Study Guide and Review

See pages439–443.

8-58-5

See pages444–449.

8-68-6

Adding and Subtracting PolynomialsConcept Summary

• To add polynomials, group like terms horizontally or write them in columnform, aligning like terms vertically.

• Subtract a polynomial by adding its additive inverse. To find the additiveinverse of a polynomial, replace each term with its additive inverse.

Find (7r2 � 9r) � (12r2 � 4).

(7r2 � 9r) � (12r2 � 4) � 7r2 � 9r � (�12r2 � 4) The additive inverse of 12r2 � 4 is �12r2 � 4.

� (7r2 � 12r2) � 9r � 4 Group like terms.

� �5r2 � 9r � 4 Add like terms.

Exercises Find each sum or difference. See Examples 1 and 2 on pages 439 and 440.

46. (2x2 � 5x � 7) � (3x3 � x2 � 2) 47. (x2 � 6xy � 7y2) � (3x2 � xy � y2)

48. (7z2 � 4) � (3z2 � 2z � 6) 49. (13m4 �7m � 10) � (8m4 � 3m � 9)

50. (11m2n2 � 4mn � 6) � (5m2n2 � 6mn � 17) 16m2n2 � 10mn � 1151. (�5p2 � 3p � 49) � (2p2 � 5p � 24) �7p2 � 2p � 2546. �3x3 � x2 � 5x � 5 47. 4x2 � 5xy � 6y2 48. 4z 2 � 2z � 10 49. 21m4 � 10m � 1


ExampleExample

ExamplesExamples

Multiplying a Polynomial by a Monomial Concept Summary

• The Distributive Property can be used to multiply a polynomial by a monomial.

1 Simplify x2(x � 2) � 3(x3 � 4x2).

x2(x � 2) � 3(x3 � 4x2) � x2(x) � x2(2) � 3(x3) � 3(4x2) Distributive Property

� x3 � 2x2 � 3x3 � 12x2 Multiply.

� 4x3 � 14x2 Combine like terms.

2 Solve x(x � 10) � x(x � 2) � 3 � 2x(x � 1) � 7.

x(x � 10) � x(x � 2) � 3 � 2x(x � 1) � 7 Original equation

x2 � 10x � x2 � 2x � 3 � 2x2 � 2x � 7 Distributive Property

2x2 � 8x � 3 � 2x2 � 2x � 7 Combine like terms.

�8x � 3 � 2x � 7 Subtract 2x2 from each side.

�10x � 3 � �7 Subtract 2x from each side.

�10x � �10 Subtract 3 from each side.

x � 1 Divide each side by �10.

Exercises Simplify. See Example 2 on page 444. 53. 10x2 � 19x � 6352. b(4b � 1) � 10b 4b2 � 9b 53. x(3x � 5) � 7(x2 � 2x � 9)

54. 8y(11y2 � 2y � 13) � 9(3y3 � 7y � 2) 55. 2x(x � y2 � 5) � 5y2(3x � 2)

61y3 � 16y2 � 167y � 18 2x2 � 17xy2 � 10x � 10y2

Solve each equation. See Example 4 on page 445.

56. m(2m � 5) � m � 2m(m � 6) � 16 2 57. 2(3w � w2) � 6 � 2w(w � 4) � 10 1�17

�




• Extra Practice, see pages 837–839.• Mixed Problem Solving, see page 860.

See pages452–457.

8-78-7

See pages458–463.

8-88-8

ExamplesExamples

ExamplesExamples

Multiplying Polynomials Concept Summary

• The FOIL method is the sum of the products of the first terms F, the outerterms O, the inner terms I, and the last terms L.

• The Distributive Property can be used to multiply any two polynomials.

1 Find (3x � 2)(x � 2).F L

F O I L(3x � 2)(x � 2) � (3x)(x) � (3x)(�2) � (2)(x) � (2)(�2) FOIL Method

I � 3x2 � 6x � 2x � 4 Multiply.

O � 3x2 � 4x � 4 Combine like terms.

2 Find (2y � 5)(4y2 � 3y � 7).

(2y � 5)(4y2 � 3y � 7)

� 2y(4y2 � 3y � 7) � 5(4y2 � 3y � 7) Distributive Property

� 8y3 � 6y2 � 14y � 20y2 � 15y � 35 Distributive Property

� 8y3 � 14y2 � 29y � 35 Combine like terms.

58. r2 � 4r � 21 59. 4a2 � 13a � 12 60. 18x2 � 0.125Exercises Find each product. See Examples 1, 2, and 4 on pages 452–454.

58. (r � 3)(r � 7) 59. (4a � 3)(a � 4) 60. (3x � 0.25)(6x � 0.5)

61. (5r � 7s)(4r � 3s) 62. (2k � 1)(k2 � 7k � 9) 63. (4p � 3)(3p2 � p � 2)

20r 2 � 13rs � 21s2 2k3 � 15k2 � 11k � 9 12p3 � 13p2 � 11p � 6

Special ProductsConcept Summary

• Square of a Sum: (a � b)2 � a2 � 2ab � b2

• Square of a Difference: (a � b)2 � a2 � 2ab � b2

• Product of a Sum and a Difference: (a � b)(a � b) � (a � b)(a � b) � a2 � b2

1 Find (r � 5)2.


(r � 5)2 � r2 � 2(r)(5) � 52 a = r and b = 5

� r2 � 10r � 25 Simplify.

2 Find (2c � 9)(2c � 9).


(2c � 9)(2c � 9) � 2c2 � 92 a = 2c and b = 9

� 4c2 � 81 Simplify.

Exercises Find each product. See Examples 1, 2, and 4 on pages 459 and 461.

64. (x � 6)(x � 6) x2 � 36 65. (4x � 7)2 66. (8x � 5)2

67. (5x � 3y)(5x � 3y) 68. (6a � 5b)2 69. (3m � 4n)2

25x2 � 9y2 36a2 � 60ab � 25b2 9m2 � 24mn � 16n2

65. 16x2 � 56x � 49 66. 64x2 � 80x � 25


Practice Test

Chapter 8 Practice Test 469

Vocabulary and ConceptsVocabulary and Concepts

Skills and ApplicationsSkills and Applications

1. Explain why (42)(43) 165. (42)(43) � (4 � 4)(4 � 4 � 4) � 4 � 4 � 4 � 4 � 4 or 45 and 165 � 45.

2. Write �15

� using a negative exponent. 5– 1

3. Define and give an example of a monomial. Sample answer: A monomial is a number, variable, orproduct of numbers and variables; 6x2.

Simplify. Assume that no denominator is equal to zero. 5. �48a3b5c4. (a2b4)(a3b5) a5b9 5. (�12abc)(4a2b4) 6. ��

35

�m�2

�295�m2 7. (�3a)4(a5b)2 81a14b2

8. (�5a2)(�6b3)2 9. �mm

3nn

4

2� �mn2

2� 10. �693aa

2b4bc2

c� �

7ca2� 11. �

(438aab

2

3bc2c)

5

2� �31b6c

5�


12. 46,300 4.63 104 13. 0.003892 3.892 10�3 14. 284 � 103 2.84 105 15. 52.8 � 10�9

Evaluate. Express each result in scientific notation and standard notation.

16. (3 � 103)(2 � 104) 17. �134..272

��

1100�

�

3

4� 4.6 10�1; 0.46 18. (15 � 10�7)(3.1 � 104)

19. SPACE EXPLORATION A space probe that is 2.85 � 109 miles away from Earth sends radio signals to NASA. If the radio signals travel at the speed of light (1.86 � 105 miles per second), how long will it take the signals to reach NASA? about 1.53 104 s or 4.25 h

Find the degree of each polynomial. Then arrange the terms so that thepowers of y are in descending order.

20. 2y2 � 8y4 � 9y 4; 8y4 � 2y2 � 9y 21. 5xy � 7 � 2y4 � x2y3 5; 2y4 � x2y3 � 5xy � 7


22. (5a � 3a2 � 7a3) � (2a � 8a2 � 4) 23. (x3 � 3x2y � 4xy2 � y3) � (7x3 � x2y � 9xy2 � y3)

24. GEOMETRY The measures of two sides of a triangle aregiven. If the perimeter is represented by 11x2 � 29x � 10, find the measure of the third side. 5x2 � 23x � 23

Simplify.

25. (h � 5)2 h2 � 10h � 25 26. (4x � y)(4x � y) 16x2 � y2

27. 3x2y3(2x � xy2) 6x3y3 � 3x3y5 28. (2a2b � b2)2 4a4b2 � 4a2b3 � b4

29. (4m � 3n)(2m � 5n) 8m2 � 14mn � 15n2 30. (2c � 5)(3c2 � 4c � 2) 6c3 � 7c2 � 16c � 10


31. 2x(x � 3) � 2(x2 � 7) � 2 2 32. 3a(a2 � 5) � 11 � a(3a2 � 4) 1

33. STANDARDIZED TEST PRACTICE If x2 � 2xy � y2 � 8, find 3(x � y)2. C2 4

24 cannot be determinedDC

BA

5x 2 � 13x � 24

x 2 � 7x � 9

www.algebra1.com/chapter_test

�180a2b6

5.28 10�8

6 107; 60,000,000 4.65 10�2; 0.0465

7a � 5a2 � 7a3 � 4 �6x3 � 4x2y � 13xy2

Chapter 8 Practice Test 469

Introduction Error analysis shows common mistakes that happen when performingan operation. An example of a error when multiplying like bases is 43 � 44 � 167.Actually, 43 � 44 � 47. The error occurred by multiplying the bases while adding the exponents. The base should stay the same while adding the exponent.Ask Students From the material in this chapter, find a problem that occurs oftenand write an error analysis for it. Describe the situation, give the correct method forthat example, and write a paragraph about it. Place this in your portfolio.

Portfolio Suggestion

Assessment Options

Vocabulary Test A vocabularytest/review for Chapter 8 can befound on p. 516 of the Chapter 8Resource Masters.

Chapter Tests There are sixChapter 8 Tests and an Open-Ended Assessment task availablein the Chapter 8 Resource Masters.

Open-Ended AssessmentPerformance tasks for Chapter 8can be found on p. 515 of theChapter 8 Resource Masters. Asample scoring rubric for thesetasks appears on p. A31.


This networkable software hasthree modules for assessment.

• Worksheet Builder to makeworksheets and tests.

• Student Module to take testson-screen.

• Management System to keepstudent records.

Chapter 8 TestsForm Type Level Pages

1 MC basic 503–504

2A MC average 505–506

2B MC average 507–508

2C FR average 509–510

2D FR average 511–512

3 FR advanced 513–514

MC = multiple-choice questionsFR = free-response questions



Standardized Test PracticeStudent Record Sheet (Use with pages 470–471 of the Student Edition.)

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7 9

2 5 8 10

3 6

Solve the problem and write your answer in the blank.

For Questions 11 and 13, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.

11 (grid in) 11 13

12

13 (grid in)

14

15

16

Select the best answer from the choices given and fill in the corresponding oval.

17 21

18 22

19 23

20

Record your answers for Question 24 on the back of this paper.

DCBA

DCBADCBA

DCBADCBA

DCBADCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

DCBADCBA

DCBADCBADCBADCBA

DCBADCBADCBADCBA

NAME DATE PERIOD

88

An

swer

s

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 4 Open-EndedPart 4 Open-Ended

Part 1 Multiple ChoicePart 1 Multiple Choice

Standardized Test PracticeStudent Recording Sheet, p. A1

Additional Practice

See pp. 521–522 in the Chapter 8Resource Masters for additionalstandardized test practice.

Teaching Tip Urge students tocheck their answer to Question 7 bysubstituting it back into the originalquestion.

Record your answers on the answer sheetprovided by your teacher or on a sheet ofpaper.

1. A basketball team scored the following pointsduring the first five games of the season: 70, 65, 75, 70, 80. During the sixth game, they scored only 30 points. Which of thesemeasures changed the most as a result of thesixth game? (Lessons 2-2 and 2-5) A

mean

median

mode

They all changed the same amount.

2. A machine produces metal bottle caps. Thenumber of caps it produces is proportional tothe number of minutes the machine operates.The machine produces 2100 caps in 60 minutes.How many minutes would it take the machineto produce 5600 caps? (Lesson 2-6) D

35 58.3 93.3 160

3. The odometer on Juliana’s car read 20,542 mileswhen she started a trip. After 4 hours of driving,the odometer read 20,750 miles. Whichequation can be used to find r, her averagerate of speed for the 4 hours? (Lesson 3-1) D

r � 20,750 � 20,542

r � 4(20,750 � 20,542)

r � �20,

4750�

r � �20,750 �

420,542�

4. Which equation bestdescribes the graph?(Lesson 5-4) A

y � ��15

�x � 1

y � �5x � 1

y � �15

�x � 5

y � �5x � 5

5. Which equation represents the line thatpasses through the point at (�1, 4) and has a slope of �2? (Lesson 5-5) B

y � �2x � 2 y � �2x � 2

y � �2x � 6 y � �2x � 7

6. Mr. Puram is planning an addition to theschool library. The budget is $7500. Eachbookcase costs $125, and each set of table andchairs costs $550. If he buys 4 sets of tables andchairs, which inequality shows the number ofbookcases b he can buy? (Lesson 6-6) A

4(550) � 125b � 7500

125b � 7500

4(550 � 125)b � 7500

4(125) � 550b � 7500

7. Sophia and Allie went shopping and spent$122 altogether. Sophia spent $25 less thantwice as much as Allie. How much did Alliespend? (Lesson 7-2) B

$39 $49 $53 $73

8. The product of 2x3 and 4x4 is (Lesson 8-1) D

8x12. 6x12. 6x7. 8x7.

9. If 0.00037 is expressed as 3.7 � 10n, what isthe value of n? (Lesson 8-3) B

�5 �4 4 5

10. When x2 � 2x � 1 is subtracted from 3x2 � 4x � 5, the result will be (Lesson 8-5) A

2x2 � 2x � 4. 2x2 � 6x � 4.

3x2 � 6x � 6. 4x2 � 6x � 6.DC

BA

DCBA

DCBA

DCBA

D

C

B

A

DC

BA

D

C

B

A

y

xO

D

C

B

A

DCBA

D

C

B

A

Part 1 Multiple Choice


Test-Taking TipQuestion 5 When you write an equation, checkthat the given values make a true statement. Forexample, in Question 5, substitute the values of thecoordinates (�1, 4) into your equation to check.


These two pages contain practicequestions in the various formatsthat can be found on the mostfrequently given standardizedtests.

A practice answer sheet for thesetwo pages can be found on p. A1of the Chapter 8 Resource Masters.

Log On for Test Practice The Princeton Review offersadditional test-taking tips and

practice problems at their web site. Visitwww.princetonreview.com orwww.review.com


Special banks of standardized testquestions similar to those on the SAT,ACT, TIMSS 8, NAEP 8, and Algebra 1End-of-Course tests can be found onthis CD-ROM.

Log On for Test Practice The Princeton Review offersadditional test-taking tips and

practice problems at their web site. Visitwww.princetonreview.com orwww.review.com


Special banks of standardized testquestions similar to those on the SAT,ACT, TIMSS 8, NAEP 8, and Algebra 1End-of-Course tests can be found onthis CD-ROM.

http://www.princetonreview.com

http://www.review.com

Evaluating Open-EndedAssessment Questions

Open-Ended Assessment ques-tions are graded by using a multi-level rubric that guides you inassessing a student’s knowledgeof a particular concept.

Goal: Write polynomial expres-sions to describe surface area andvolume of a prism.

Sample Scoring Rubric: The fol-lowing rubric is a sample scoringdevice. You may wish to add moredetail to this sample to meet yourindividual scoring needs.

Answers

14.

24c. 3m3 � 4m2 � 3m � 12

y

xO

3x � y � 2

Record your answers on the answer sheetprovided by your teacher or on a sheet ofpaper.

11. Find the 15th term in the arithmetic sequence�20, �11, �2, 7, … . (Lesson 4-7) 106

12. Write a function that includes all of theordered pairs in the table. (Lesson 4-8)

13. Find the y-intercept of the line representedby 3x � 2y � 8 � 0. (Lesson 5-4) 4

14. Graph the solution of the linear inequality3x � y � 2. (Lesson 6-6) See margin.

15. Let P � 3x2 � 2x � 1 and Q � �x2 � 2x � 2.Find P � Q. (Lesson 8-5) 2x2 � 3

16. Find (x2 � 1)(x � 3). (Lesson 8-7)x3 � 3x2 � x � 3

Compare the quantity in Column A and thequantity in Column B. Then determinewhether:

the quantity in Column A is greater,

the quantity in Column B is greater,

the two quantities are equal, or

the relationship cannot be determinedfrom the information given.

17.

A (Lesson 4-3)

18.

A (Lesson 6-3)

19.

B (Lesson 7-3)

20.

C (Lesson 8-2)

21.

A (Lesson 8-3)

22.

C (Lesson 8-4)

23. m2 � n2 � 10 and mn � �6

B (Lesson 8-8)

Record your answers on a sheet of paper.Show your work.

24. Use the rectangular prism below to solve the following problems. (Lessons 8-1 and 8-7)

a. Write a polynomial expression thatrepresents the surface area of the top ofthe prism. 3m2 � 3

b. Write a polynomial expression thatrepresents the surface area of the front ofthe prism. 3m2 � 9m � 12

c. Write a polynomial expression thatrepresents the volume of the prism.

d. If m � 2 centimeters, then what is thevolume of the prism? 54 cm3

3m � 3m � 1

m � 4

y

xO

A

B

D

C

B

A

Part 2 Short Response/Grid In

Part 4 Open Ended

Part 3 Quantitative Comparison

www.algebra1.com/standardized_test Chapter 8 Standardized Test Practice 471

Aligned and verified by

Column A Column B

4x � 10 20 ��6(x

8� 1)� 3

�2b

4

�

b

3

cc2

� �20

1b08bc

4

�1�

the x value in the x value inthe solution of the solution of

x � 3y � 2 and 3x � 8y � 6 andx � 3y � 0 x � 8y � 2

5.01 � 10�2 50.1 � 10�4

the domain of the range ofpoint A point B

�3 �1 1 3 4

12 4 �4 �12 �16

x

yy � �4x or f(x) � �4x

the degree of the degree ofx2 � 5 � 6x � 13x3 10 � y � 2y2 � 4y3

(m � n)2 (m � n)2

c. See margin.

Chapter 8 Standardized Test Practice 471

Score Criteria4 A correct solution that is supported

by well-developed, accurateexplanations

3 A generally correct solution, butmay contain minor flaws inreasoning or computation

2 A partially correct interpretationand/or solution to the problem

1 A correct solution with nosupporting evidence or explanation

0 An incorrect solution indicating nomathematical understanding of the concept or task, or no solution is given


Pages 413–415, Lesson 8-1

61.

62. 63.

Page 416, Follow-Up of Lesson 8-1

2. Sample answer:

Prism Dimensions S.A. Vol. S.A. Ratio Vol. Ratio

Original 4 by 6 by 9 228 216 — —

A 8 by 12 by 18 912 1728 � 4 � 8

B 12 by 18 by 27 2052 5832 � 9 � 27

6. Yes, the conjectures hold. If the ratio of the dimensionsof two cylinders is a, then the ratio of the surface areasis a2 and the ratio of the volumes is a3.


51. You can compare pH levels by finding the ratio of onepH level to another written in terms of the

concentration c of hydrogen ions, c � � �pH.

Answers should include the following.

• Sample answer: To compare a pH of 8 with a pH of 9requires simplifying the quotient of powers,

� � ��110��8 � 9

� ��110��1

� Negative Exponent Property

� 10

Thus, a pH of 8 is ten times more acidic than a pH of 9.


60a. always

(a � 10p) � ap � (10n)p Product of Powers

� ap � 10np Power of a Product

60b. Sometimes; ap � 10np is only in scientific notation if 1 � ap � 10. Counterexample: (5 �103)2 � 52 � 106

or 25 � 106, but 25 � 106 is not in scientific notationsince 25 is greater than 10.

61. Astronomers work with very large numbers such asthe masses of planets. Scientific notation allows themto more easily perform calculations with thesenumbers. Answers should include the following.

•

• Scientific notation allows you to fit numbers such asthese into a smaller table. It allows you to comparelarge values quickly by comparing the powers of 10instead of counting zeros to find place value. Forcomputation, scientific notation allows you work withfewer place values and to express your answers in acompact form.

Page 431, Preview of Lesson 8-4

1.

2.

3.

4.

x x x xx 2

1 1 1

�xx 2 x 2x 2

x x x x�1 �1 �1 �1

�x 2�x 2

Planet Mass (kg)

Mercury 330,000,000,000,000,000,000,000

Venus 4,870,000,000,000,000,000,000,000

Earth 5,970,000,000,000,000,000,000,000

Mars 642,000,000,000,000,000,000,000

Jupiter 1,900,000,000,000,000,000,000,000,000

Saturn 569,000,000,000,000,000,000,000,000

Uranus 86,800,000,000,000,000,000,000,000

Neptune 102,000,000,000,000,000,000,000,000

Pluto 12,700,000,000,000,000,000,000

1�

��110��1

��110��8�

��110��9��110��8�

��110��9

1�10

5832�216

2052�228

1728�216

912�228

y

xO

x � �2

y � x � 3

y

xO

y � 2x � 1

y � x � 2

y

xO

y � 2x � 2

y � �x � 1

471A Chapter 8 Additional Answers

Addit

ion

al

An

sw

ers

for

Ch

apte

r 8


58. A polynomial model of a set of data can be used topredict future trends in data. Answers should includethe following.

The polynomial function models the data exactly for thefirst 3 values of t, and then closely for the next 3 values.

• Someone might point to this model as evidence thatthe time people spend playing video games is on therise. This model may assist video game manufacturersin predicting production needs.

Page 438, Preview of Lesson 8-5

7. Method from Activity 2:

You need to add zero pairs so that you can remove 2green x tiles and 3 yellow 1 tiles.

Method from Activity 3:

You remove all zero pairs to find the difference insimplest form.

Page 446–449, Lesson 8-6

63. Answers should include the following.• The product of a monomial and a polynomial can be

modeled using an area model. The area of the figureshown at the beginning of the lesson is the productof its length 2x and width (x � 3). This product is2x (x � 3), which when the Distributive Property isapplied becomes 2x (x ) � 2x (3) or 2x2 � 6x. This isthe same result obtained when the areas of thealgebra tiles are added together.

• Sample answer: (3x )(2x � 1)

(3x)(2x � 1) � (3x)(2x) � (3x)(1)

� 6x2 � 3x


47.

Area of rectangle � (a � b)(a � b)

or

Area of a trapezoid � (height)(base 1 � base 2)

A1 � (a � b)(a � b)

A2 � (a � b)(a � b)

Total area of shaded region

� � (a � b)(a � b)� � � (a � b)(a � b)�� (a � b)(a � b)

1�2

1�2

1�2

1�2

1�2

aA1

A2b

b

a – b

a – b

a

a � b a � b

a � b

b a

a b

a � b

a – b

a – b

a

a

b

b

x x 2 xx 2

x 2 xx 2

x 2 xx 2

x

x

x

x 1

�x �x�1 �1 �1

�x �x �xx 2x 2

1

Opposite of 2x � 3

2x 2 � 5x � 2

�x �x �x

x �x x �x

x 2

x 2

�1 �1

1 1 1

2x 2 � 5x � 2

t H Actual Data Values

0 19 19

1 19 19

2 22 22

3 23.5 24

4 25 26

5 34 36

Chapter 8 Additional Answers 471B

Additio

nal A

nsw

ers

for C

hapte

r 8

notes polynomials and nonlinear introduction functions · • simplify expressions containing...

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