Algebra Chapter 11

11-1 11-2 11-311-4 11-5 11-6

11-7 11-8 (dropped)

Chapter Review

11-1 Investments and Polynomials

Objectives: Add and subtract polynomials

Translate investment situations into polynomials

When amounts are invested periodically and earn interest from the time of investment, the total value can be represented by a polynomial.

11.1

Definitions:

Polynomial in x – is a sum of multiples of powers of x. For example: -2x3 + 3x2 + 2x + 6

Standard form for polynomials are polynomials written in decreasing powers of x.

Scale Factor: the amount of increase to be multiplied - interest rate

11.1

1) As a New Year's resolution, Bert has decided to deposit $100 in a savings account every January 2nd. The account yields 3% interest annually. How much will his savings be worth when he makes his fourth deposit?

Jan 2nd – 1st deposit = $100

Jan 2nd – 2nd deposit 100(1.03) + 100 = $203

Jan 2nd – 3rd deposit 100(1.03)2 + 100(1.03) +100 = $309.09

Jan 2nd – 4th deposit 100(1.03)3 + 100(1.03)2 + 100(1.03) +100 = $418.36

11.1

2) Janice has a savings account that has a scale factor of x. She makes deposits at regular yearly intervals. The first year she deposits $800, the second year $300, the third year $450, and the fourth year $775. What is her balance immediately after the fourth deposit?

Jan 2nd – 1st deposit = 800

Jan 2nd – 2nd deposit = 800x + 300

Jan 2nd – 3rd deposit = 800x2 + 300x + 450

Jan 2nd – 4th deposit = 800x3 + 300x2 + 450x + 775

11.1

3) Which is more advantageous, to invest $50 per year for four years or to invest $100 in the first year and $100 in the fourth year? In both instances, the money earns 3% interest a year.

$50 per year – 4th deposit = 50(1.03)3 + 50(1.03)2 + 50(1.03) + 50 = $209.18

$100 1st & 4th – 4th deposit = 100(1.03)3 + 100 = $209.27

You’ll make $0.09 more depositing $100 in the 1st and 4th years

11.1

4) Simplify

a) 4x4 + 2x3 - x - 4x(x2 - 4x + 2)

b) 5x4 + 2x2 - x + (?) = x2 - 4x + 2

4x4 + 2x3 - x - 4x3 + 16x2 - 8x

4x4 + 2x3 - 4x3 + 16x2 - x - 8x

4x4 - 2x3 + 16x2 - 9x

- 5x4 -5x4 Get rid of the x4 term

- 2x2 = (-5x4) +x2 -2x2 - 4x + 2 Decrease x2 term by 2x2

+ x = (-5x4-x2) -4x + x + 2Decrease x term by 3

5x4 + 2x2 - x + (-5x4-x2-3x+2) = x2 - 4x + 2 Add 2

Distribute -4x first

Use Commutative Property – get like terms together.

Simplify & standard form

11.1

___?__ = -5x4-x2-3x+2

11-2 Classifying Polynomials

Objective: Classify polynomials by their degree or number of terms. Add and subtract polynomials.

Big Idea: Polynomials are classified by their number of terms and by their degree.

Goal: Understand the basic terminology of polynomials and the classification of polynomials by the number of terms or by their degree.

11.2

Term Review

5x³yvariable

coefficient

exponent

•Coefficients of 1 are implied: x³ = 1∙x³•Exponents of 1 are implied.•An expression is considered to be simpler when it is written as x rather than 1x or . •If there is no variable, the term is called a constant. It’s degree is zero. 5 =•To find the total degree: add the exponents of all the variables

13 3y y

1x

05x

What is a polynomial?• An expression that has no operations other than

addition, subtraction, and multiplication by or of the variables. Every exponent must be a non-negative integer in a polynomial.

Not polynomials:• Fractional Exponents ex. Square Roots• Absolute Values• Terms divided by a Variable• Terms with Negative Exponents (these are

actually terms divided by a variable)

26 12x x

x

Polynomials vs. Not Polynomials

|10-2y|

66x

825x

• 4x• 5• 3x²-5x³+2x-4• 4y-3• 3z³+6•

•

214

2x

26 12

6

x x

Vocabulary:

A. Classifying Polynomials by # of Terms

1) Monomial - an expression that can be written as a real number, a variable, or a product of a real number and one or more variables with non-negative exponents. ex. 6, x, 6xyz

2) Polynomial - an algebraic expression that is either a monomial or a sum of monomials

3) Binomial - a sum of two monomials ex. 5x + 3

4) Trinomial - a sum of three monomials

ex. 5xy + 3x + 5

11.2

B. Classifying Polynomials by Degree:

5) Degree of a Monomial - the sum of the exponents of the variables in the monomial

Ex. 6xyz degree 3; -5x³ degree 3

6) Degree of a Polynomial - the highest degree of any of its terms after the polynomial has been simplified.

7) Linear - a polynomial of degree 1. ex. 3x + 1

8) Quadratic - a polynomial of degree 2. ex. 4x²

9) Cubic – a polynomial of degree 3 ex. 3x³ - 5

11.2

Monomial/Binomials/ Trinomials

Name Number of Terms

Example

Monomial 1 -5x³

Binomial 2 4x²-5x

Trinomial 3 4x²+2x-3

#-degree 4+ 7 34 3 2x x x

Degree Summary

Degree Name Example

First Linear 4x+5

Second Quadratic 5x²-4x+6

Third Cubic x³+6x

Fourth Quartic

Fifth Quintic

6+ No special name#-degree

The degree of a polynomial with one variable is the exponent of the highest power of that variable.

4 3 2 5x x x

5 32 2 5x x x

7 34 2 5x x x

Classifying Polynomials

• Can be done by both degree and number of terms

• Standard: State the degree first and then the number of terms

Classification of Polynomial Examples:

State the degree and the type of polynomial.

1. 5x³

2. 4x-3

3. 9-4x+2x³

4.

5. 4x

4 3 2 5x x x

6. 3x²-5x³+2x-4

7. 4y-3

8. 3z³+6

9. 25x²-100

10. 3x²-5x³+2x-4

11. 5

Write an example of each type of polynomial:

1. Linear binomial

2. Cubic trinomial

3. A 4th degree monomial

4. A quadratic trinomial

Descending OrderTerms in a polynomial are listed in descending order of the exponents on the variable. As you go from left to right, the exponents go down in value.

This is considered STANDARD FORM.

NOTE: The value of the coefficient is not considered only the exponent value!

Ex A. Write in Standard Form: 1 + 3y - 4y2 - 5y3

B. Write in Descending Form: -3 - 5x³ + y -10y²

-5y3 - 4y2 + 3y + 1

– 5x3 – 10y2 + y - 3

Ascending OrderTerms in a polynomial are listed in ascending order of the exponents on the variable. As you go from left to right, the exponents go up in value.

Ex. Arrange in this polynomial in ascending order: 9y - 5y² - 4y³+ 1

1+ 9y - 5y2 - 4y3 + 1

Arranging terms in ascending/ descending order

• Simplify the polynomial first

• Commute the terms so that the exponents either increase or decrease from term to term.

• Always remember that the negative sign goes along with the term it proceeds.

• Remember if there is no exponent for a variable the exponent is a 1.

Arrange this polynomial in both ascending and descending order:

5 2 3 4 5 700 3 12 36x x x x x

Standard form for more than one variable:There is no standard form, however, sometimes one

variable is picked and the polynomial is written in decreasing powers of that variable.

Example: a. Write in Standard Form as a polynomial in p.- q + 3p - 4p2 – p3q2 + p2q3

b. Write in Standard Form as a polynomial in q. p - q + 3p - 4p2 – p3q2 + p2q3

– p3q2 + p2q3 - 4p2 + 3p - q

p2q3 - p3q2 - 4p2 + 3p - q

Adding and Subtracting Monomials Review

• Only add/subtract like terms. Remind me, what are like terms?

• Add coefficients keep exponent.

• Ex: 5x²-3x+4x²

Multiplying Monomials Review

• Multiply like bases.

• Add exponents, keep bases.

• Multiply coefficients

• Ex: (4xy)(3xy³)(-2x²)

Simplify.

1. 3x2 + 5x2

2. 4x3 - 2x2

3. 2x6 - (4x6+7x6-9x6)

4. (11x)(-3x)

5. ab + ba

More Practice Examples:

1) Tell if the expression is a monomial. If so, identify its degree. If not tell why.

a. 15x2 b. 156

c. d. ¾ x3

e. x3y4 f. 2x + 15x2

3

5

x

Yes – deg: 2 Yes – deg: 0

No – neg exponent Yes – deg: 3

Yes: deg: 7 No - binomial

11.2

2) Give the degree of each polynomial

a. 1 + 3y + 4y2 + 5y3

b. p - q + 3p - 4p2 - p2q2

c. 4x4y-1 * + 7xy

33

2

y

x

Polynomial degree = 3

Polynomial degree = 4

= -6x3y2 + 7xy

Polynomial degree = 5

11.2

List the following terms in ascending order.

5 2 3

5 10

3. 4 3 10 40

4. 2 5 11 5

x x x x

x x x

5)

a) Write a monomial with one variable whose degree is 5.

b) Write a monomial with two variables whose degree is 5.

c) Write a trinomial with degree 5.

Sample answer : ½x5

Sample answer : -4x3y2

Sample answer : -4x3y2 +xy + 7

11.2

11-3 Multiplying a Polynomial by a Monomial

Objective: Multiply a polynomial by a monomial.

Represent areas of figures with polynomials.

Big Idea: To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial and add the products.

Goal: Apply the distributive property to multiply a polynomial by a monomial using area models to picture the porducts.

11.3

1) Give two equivalent expressions for the area pictured below. One is simplified. X + X + X = 3X

(3x)X

+

X

+

1+1+1+1

= 2X + 4

(2x + 4) = distribute

(3x)(2x) + (3x)(4) =

X2 + X2 + X2 +

X2 + X2 + X2 +

x + x + x +x + x + x +x + x + x +x + x + x = (3x)(2x+4) = 6x2 + 12x

11.3

2) Multiply k4(k2 - 16km)11.3

= k6 - 16k5m

3) Multiply -5y(y3 - 6y2 + 2y + 6)11.3

(-5y)(y3) – (-5y)(6y2) + (-5y)(2y) + (-5y)(6) Distribute

-5y4 + 30y3 - 10y2 - 30y Multiply

4) Simplify

3x(x2 – 5) – (3x2 + 4x – 5) + 2(-2x2 – 4x)

3x3 - 15x - 3x2 - 4x + 5 - 4x2 - 8x Distribute

3x3 - 3x2 - 4x2 - 15x - 4x - 8x + 5 Commutative prop

3x3 - 7x2 - 27x + 5 Simplify

11.3

5) Draw boxes that would represent:

(2x)(3x+2) = 6x2 + 4x

11.3

x

+

x

x + x + x + 1 + 1

11-4 Common Monomial Factoring

Understand factoring as the reverse process of multiplication, concentrating on common monomial factors and their applications to the division of a polynomial by a monomial.

11.4

Definitions:

Factoring – the expression of a single monomial as the product of two or more factors.

Trivial Factors – 1 and itself.

Greatest Common Factor – the product of the gcf of the coefficients and the gcf of the variables.

Factorization – the result of factoring a polynomial.

Prime polynomials – monomials and polynomials that cannot be factored into polynomials of a lower degree.

Complete factorization – When there are no common numerical factors in the terms of any of the prime polynomials. Ex 6x+12 factored completely is 6(x+2)

11.4

Unique Factorization Theorem for Unique Factorization Theorem for PolynomialsPolynomialsEvery polynomial can be represented as a

product of prime polynomials in exactly one way, disregarding order and integer multiples.

Factoring is the process of expressing a given number or expression as a product.

The factored expression is always equivalent to the original polynomial.

11.4

Steps for Factoring

1.Find GCF for all terms.

2.Divide all terms by GCF.

3.Write the answer as a product of the GCF and the quantity of the remaining factors.

NOTE: No terms should be lost in this process!

The number of terms in the parenthesis should be the same as the original polynomial!

11.4

1) What are the factors of 8x3?

Factor the coefficient8

1, 82, 4

Factor the variable(s)x3

xx2

x3

Use all the individual factors and then combine them.1, 2, 4, 8, x, x2, x3, 8x, 8x2, 2x, 2x2, 2x3, 4x, 4x2, 4x3

2) What is the greatest common factor between 8x3 and 12xy2

4x

11.4

3) Factor 4x5 + 12x3 + 8x

GCF of all 3 terms is 4x

= 4x (x4 + 3x2 + 2) Simplify all terms inside the parenthesis.

Factor/Divide the GCF from all terms.

5 34 12 84

4 4 4

x x xx

x x x

11.4

4) Simplify 3 224 6

12

y y

y

26 (4 )

6 (2)

y y y

y

Find GCF of ALL terms top & bottomAnd factor it out….

24

2

y ySimplify fraction : Cancel

Final answer

11.4

5) Illustrate the factorization of 4x2 + 12x by drawing a rectangle whose sides are the factors.

= 4x (x + 3)x + 1+1+1

x+x+x+x

11.4

11-5 Multiplying Polynomials

Objectives: Multiply polynomials having two or more terms.

Represent areas and volumes of figures with polynomials.

11.5

Property:

The Extended Distributive Property - To multiply two sums, multiply each term in the first sum by each term in the second sum.

11.5

Property: Multiplying two binomials

The FOIL algorithm:

First Outside Inside Last

(a + b)(c + d) =

11.5

+ bdac + ad + bc

1) Multiply (3n + 10)(7n + 2)11.5

= (3n)(7n) + (3n)(2) + (10)(7n) + (10)(2)

= 21n2 + 6n + 70n + 20 multiply

= 21n2 + 76n + 20 simplify

(n - 5)(2n2 - 3n + 7)

11.52) Multiply a binomial by a trinomial – remember: The Extended Distributive Property – To multiply two sums, multiply each term in the first sum by each term in the second sum.

= (n)(2n2) - (n)(3n) + (n)(7) - (5)(2n2) + (5)(3n) - (5)(7)

= 2n3 - 3n2 + 7n - 10n2 + 15n - 35 multiply

= 2n3 - 13n2 + 22n - 35 simplify

(w2 + 4w + 6)(w2 + w + 1)

11.53) Multiply a trinomial by a trinomial

= w2(w2) + w2(w) + w2(1) + 4w(w2) + 4w(w) + 4w(1) + 6(w2) + 6(w) + 6(1)

= w4 + w3 + w2 + 4w3 + 4w2 + 4w + 6w2 + 6w + 6

= w4 + 5w3 + 11w2 + 10w + 6 simplify

multiply

Remember standard form

4) (2a + 10b + 1)(4a + b+ 1)11.5

= 2a(4a) + 2a(b) + 2a + 10b(4a) + 10b(b) + 10b + 4a + b + 1

= 8a2 + 2ab + 2a + 40ab + 10b2 + 10b + 4a + b + 1

= 8a2 + 6a + 42ab + 10b2 + 11b + 1 simplify

5) a. Express the area as the sum of the tiles.

b. Express the area as the length * width

c. What equality is shown?

11.5

x + x 1+1+1

x+x+x+x+1

= (2x + 3)(4x + 1)

= 8x2 + 14x + 3

= 8x2 + 14x + 3 = (2x+3)(4x+1)

x2 x2 x x x

x2 x2 x x x

x2 x2 x x x

x2 x2 x x x

x x 1 1 1

11-6 Special Binomial Products

Objectives: Apply two patterns of binomial multiplication, the square of a binomial and the difference of squares, to do arithmetic multiplication mentally and to illustrate how a knowledge of algebra can contribute to increased arithmetic proficiency.

• Multiply two binomials• Expand squares of binomials• Represent the square of a binomial as an area

11.6

Properties:

1) Perfect Square Patterns: For all numbers a and b,

(a+b)2 = (a+b)(a+b) = a2 + 2ab + b2

(a-b)2 = (a-b)(a-b) = a2 - 2ab + b2

2) Difference of Two Squares Pattern: For all numbers a and b,

(a+b)(a-b) = a2 - b2

11.6

1) The length of the side of a square is y + 7.

a. Write the area of the square in expanded form.

b. Draw the square and show how the expanded form relates to the figure.

11.6

Remember: (a+b)2 = a2 + 2ab + b2

(y+7)2 = y2 + 2(y)(7) + 72

(y+7)2 = y2 + 14y + 49

y +1+1+1+1+1+1+1

y +1+1+1+1+1+1+1

2) Expand (2n-5)211.6

Remember: (a-b)2 = a2 – 2ab + b2

(2n-5)2 = (2n)2 – 2(2n)(5) + (5)2

(2n-5)2 = 4n2 – 20n + 25

3) Multiply (10n - 7)(10n + 7)

Remember: (a+b)(a-b) = a2 - b2

(10n+7)(10n-7) = (10n)2 - 72

(10n+7)(10n-7) = 100n2 - 49

4) An 8" by 8" square photograph is to be surrounded by a square mat with width w. Sketch the photo and mat. Express the area of the mat that shows as a product of 2 binomials.

Let w = mat width (and length)

Mat Area = w2 - 82

Mat Area = (w+8)(w-8)

11.6

5) Compute 512 in your head. 11.6

512 = (50 + 1)2

Remember: (a+b)2 = a2 + 2ab + b2

512 = (50+1)2 = 502 + 2(50)(1) + 12

= 2500 + 100 + 1

= 2601

Review for Final• Solve for x

3x + 4y – z = 4x + 7y

• Solve for y

3x + 4y – z = 4x + 7y

11-7 Permutations

Objectives :

Find the number of permutations of objects without replacement.

Understand factorial notation.

11.7

• Permutation: An arrangement where order is important. Example P(14,4) = 14 • 13 • 12 • 11

• Factorial: n! means the product of all counting numbers from n down to 1. Example 6! = 6 • 5 • 4 • 3 • 2 • 1 = 720 P(6,6) = 6!

11.7

Ex 1) There are 10 players on a softball team. In how many ways can the manager choose three players for first, second, and third base?

10 9 8 720x x =

number of possible players for first base

number of possible

players for second base

number of possible

players for third base

total number of possible

waysx x =

Answer: There are 720 different ways the manager can pick players for first, second, and third base.

= Permutations of 10 players chosen 3 at a time= P(10,3)

Book language: permutations chosen from 10 of length 3

11.7

Use the Fundamental Counting Principle (8-1)

(Your turn) Ex 2) There are 15 students on student council. In how many ways can Mrs. Sommers choose three students for president, vice president, and secretary?

Answer: P(15,3) = 15 · 14 · 13 = 2,730

7 things of length 5.

Ex 3) Find the value of P(7,5)

7 · 6 · 5 · 4 · 3 = 2520

11.7

90!

87!

21!

25 24 23 22 21!

90 89 88 87!

87!

Ex 4) Evaluate 5! Read “5 factorial”

5! = 5 · 4 · 3 · 2 · 1 = P(5,5)

= 120

Ex 5) Evaluate = 90·89·88 = 704880=

Ex 6) Evaluate 21!

25!= =

1

25 24 23 2

1

3036002

11.7

Ex 7) How many ways can you make an 7 digit number if you only use the digits 1-9 and you must have an even number, and no number can be used twice?

Digits to be used:

1’s column digit: 2,4,6,8, 4 available digits

1st digit: 1-9 less 1’s col digit 8 available digits

This is a permutation for the 1st 6 digits P(8,6)

7th digit must be even: there are 4 digits that would result in an even number: 2, 4, 6, 8

P(8,6) · 4 = 80,640

11.7

2nd digit: 1-9 less 2 digits 7 available digits …

Chapter 11 Review

1) Write a polynomial in standard form.

2) Write a three variable polynomial with 5 terms and a degree of 5.

Ex: 4x2 + 3x + 6

Ex: 4x2y2z + xz + xy + 3x + 6

Rev 11

3) Simplify.

a. -3x(4x2 + 2x -5)

b. (y - 4)(y + 2)

c. (3r + 1)(3r - 1)

d. (4x2 - 3x + 2) + (2x2 - 2)

-12x3 - 6x2 + 15x

y2 - 2y - 8

9r2 - 1 difference of 2 squares

Rev 11

6x2 – 3x simplify

e. (5k - 2j) - (2k - 3j + 5)

f. (4t - 2)(t4 + 3t2 + 4)

4) Factor

a. 36x2 + 12x + 6

b. ½ x2y + 4xy2 + xy

3k + j – 5 simplify

4t5 - 2t4 + 12t3 - 6t2 + 16t - 8

Rev 11

6(x2 + 2x + 1)

xy(½x + 4y + 1)

5) Know the Perfect Square Patterns and Difference of Two Squares Pattern.

1) Perfect Square Patterns:

(a+b)2 = (a+b)(a+b) = a2 + 2ab + b2

(a-b)2 = (a-b)(a-b) = a2 - 2ab + b2

2) Difference of Two Squares Pattern:

(a+b)(a-b) = a2 - b2

a) (7 + x)2

b) (2n – 6)2

c) (4z + 3a)(4z – 3a)

x2 + 14x + 49

4n2 – 24n + 36

16z2 – 9a2

Expand:

Rev 11

6) State the multiplication shown in a picture.

a.

b.

(x+2) (x+1) = x2 + 3x + 2

x · x (x+y) = x3 + x2y

Rev 11

7) Investment situation.

Each birthday from age 15 on, Joan has received $75 from her grandfather. She puts the money into a savings account with a yearly scale factor of p and does not make any withdrawals or additional deposits.

a. Write an expression for the amount Joan will have in the account on her 18th birthday.

b. If the bank pays 6% interest per year, how much will Joan have on her 18th birthday.

75p3 + 75p2 + 75p + 75

75(1.06)3 + 75(1.06)2 + 75(1.06) + 75 = $328.10

Rev 11