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C HAPTER 4: FACTORING ALGEBRAIC EXPRESSIONS

Specific Expectation Addressed in the Chapter

• Factor polynomial expressions involving common factors, trinomials, and differences of squares [e.g., 2x2 + 4x, 2x – 2y + ax – ay, x2 – x – 6, 2a2 + 11a + 5, 4x2 – 25], using a variety of tools (e.g., concrete materials, computer algebra systems, paper and pencil) and strategies (e.g., patterning). [4.1, 4.2, 4.3, 4.4, 4.5, 4.6, Chapter Task]

Prerequisite Skills Needed for the Chapter

• Determine the greatest common factor for a pair of numbers.

• Divide monomial terms.

• Expand an algebraic expression using the distributive property.

• Represent a polynomial with degree 2 using algebra tiles.

• Identify the factors in an area model.

• Sketch the parabola of a quadratic relation using its basic properties (x-intercepts, axis of symmetry, vertex).

What “big ideas” should students develop in this chapter? Students who have successfully completed the work of this chapter and who understand the essential concepts and procedures will know the following: • Factoring is the opposite of expanding. Expanding involves multiplying,

and factoring involves determining the values to multiply. • One strategy that can be used to factor an algebraic expression is to

determine the greatest common factor of the terms in the expression. For example, 5x2 + 10x – 15 can be factored as 5(x2 + 2x – 3), since 5 is the greatest common factor of the terms.

• To factor trinomials of the form ax2 + bx + c using algebra tiles, you need to form a rectangle. The factors are the dimensions of the rectangle.

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• If a quadratic expression of the form x2 + bx + c can be factored, it can be factored into two binomials: (x + r) and (x + s), where r + s = b and r × s = c.

• If the quadratic expression ax2 + bx + c, where a ≠ 1, can be factored, then the factors have the form (px + r)(qx + s), where pq = a, rs = c, and ps + rq = b.

• A polynomial of the form a2 + 2ab + b2 or a2 – 2ab + b2 is a perfect-square trinomial. • a2 + 2ab + b2 can be factored as (a + b)2. • a2 – 2ab + b2 can be factored as (a – b)2.

• A polynomial of the form a2 – b2 is a difference of squares and can be factored as (a + b)(a – b).

Chapter 4 Introduction | 127

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Chapter 4 Planning Chart

Lesson Title Lesson Goal Pacing 12 days Materials/Masters Needed

Getting Started pp. 194–197 Use concepts and skills developed prior to this chapter.

2 days grid paper; algebra tiles; ruler; Diagnostic Test

Lesson 4.1: Common Factors in Polynomials pp. 198–204

Factor algebraic expressions by dividing out the greatest common factor.

1 day algebra tiles; Lesson 4.1 Extra Practice

Lesson 4.2: Exploring the Factorization of Trinomials pp. 205–206

Discover the relationship between the coefficients and constants in a trinomial and the coefficients and constants in its factors.

1 day algebra tiles

Lesson 4.3: Factoring Quadratics: x2 + bx + c pp. 207–213

Factor quadratic expressions of the form ax2 + bx + c, where a = 1.


Lesson 4.4: Factoring Quadratics: ax2 + bx + c pp. 217–224

Factor quadratic expressions of the form ax2 + bx + c, where a ≠ 1.


Lesson 4.5: Factoring Quadratics: Special Cases pp. 225–232

Factor perfect-square trinomials and differences of squares.


Lesson 4.6: Reasoning about Factoring Polynomials pp. 233–237

Use reasoning to factor a variety of polynomials.

1 day Lesson 4.6 Extra Practice

Mid-Chapter Review: pp. 214–216 Chapter Review: pp. 238–241 Chapter Self-Test: p. 242 Curious Math: p. 232 Chapter Task: p. 243

4 days Mid-Chapter Review Extra Practice; Chapter Review Extra Practice; Chapter Test

128 | Principles of Mathematics 10: Chapter 4: Factoring Algebraic Expressions

CHAPTER OPENER

Using the Chapter Opener

Introduce the chapter by discussing the photograph on pages 192 and 193 of the Student Book. The photograph shows a criminal investigation, as indicated by the “Do Not Cross” police barrier. Ask students why working backwards is a useful strategy for figuring out what happened, step by step, to cause a given result. Students may mention that the final situation gives clues you can use to make a guess, which you then need to prove is true. Discuss how the same idea applies in mathematics, as shown by the algebraic example on page 193. Point out that students will learn how to solve for the unknown symbols and will revisit this example in the last lesson in the chapter (Lesson 4.6, question 5). Tell students that, in this chapter, they will focus on factoring algebraic expressions, which is the opposite of expanding algebraic expressions. Discuss that they have learned about expanding in Chapter 3 (and in Grade 9). Emphasize that expanding involves multiplying, while factoring involves determining the expressions to multiply.

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Chapter 4 Opener | 129

GETTING STARTED Using the Words You Need to Know Student Book Pages 194–197

Preparation and Planning

Pacing 5-10 min Words You Need to

Know 40-45 min Skills and Concepts You

Need 45-55 min Applying What You

Know

Materials grid paper algebra tiles ruler

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Students might read the terms and select the mathematical expression for each, look at the mathematical expression and search for the terms, eliminate choices by matching in an order according to what they are sure of, use a combination of strategies, or develop their own strategies. After students have completed the question, ask them to provide their own example of each term.

Using the Skills and Concepts You Need Work through each example in the Student Book (or similar examples, if you would like students to see more examples), and invite students to pose questions about them. Students could model the simplifying on page 194 and the expanding on page 195 with algebra tiles. Ask students to look over the Practice questions to see if there are any they do not know how to solve. Direct attention to the Study Aid chart in the margin of the Student Book for more help. Students can work on the Practice questions in class and complete them for homework.

Using the Applying What You Know Have students work in pairs on this activity. Have them read all the information before beginning their work. Ask them to predict how many congruent squares will fit in the given diagram, which is drawn to scale. Ensure that students realize that the final geometric painting will be a larger square composed of many of these small coloured squares. After students have finished working, ask them for the total number of small squares that will be used in the painting. Discuss how to arrive at this number using logical reasoning: the painting will have 4 rows of 3 rectangles, or 12 rectangles; since each rectangle contains 12 small squares, the number of coloured squares in the painting will be 12 × 12 = 144.

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Answers to Applying What You Know A. The side length of the small squares cannot be 8 cm because 108 is not

divisible by 8. B. The side length must be a factor of 108 and 144 because a whole number

of squares must fit in that width and length, and all the squares have the same side length.

C. The side length of the largest square that can be used to divide the 108 cm by 144 cm rectangle is 36 cm.

D. The side length of the final large square painting must be a multiple of both 108 and 144 because the length and width of these rectangles must fit into the same side length of the square painting a whole number of times.

E. The side length of the smallest final square painting that can be created is 432 cm.


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Initial Assessment What You Will See Students Doing …

When students understand … If students misunderstand …

Students reason that there must be a whole number of squares in the rectangle. They understand that the number must divide evenly into both dimensions of the rectangle, so it must be a factor of both dimensions.

Students determine the greatest common factor of the rectangle’s dimensions to determine the greatest possible side length.

Students reason that the rectangles must be placed in an array to create a larger square, so the side length of the square must be a multiple of both dimensions: 108 cm and 144 cm.

Students determine the least common multiple of the rectangle’s dimensions when they are determining the least possible side length.

Students might not reach the conclusion about the factors of the rectangle’s dimensions through reasoning, but instead rely on the diagram to figure out how many squares will fit inside the rectangle.

Students may not calculate the answer by determining the greatest common factor.

Students may not make the connection between the dimensions of the rectangle and the dimensions of the larger square.

Students cannot calculate the answer by determining the least common multiple.

Chapter 4 Getting Started | 131

4.1 COMMON FACTORS IN POLYNOMIALS Lesson at a Glance

Prerequisite Skills/Concepts • Determine the greatest common factor for a pair of numbers. • Divide monomial terms. • Expand an algebraic expression using the distributive property. • Represent a polynomial with degree 2 using algebra tiles.

Specific Expectation • Factor polynomial expressions involving common factors, [trinomials,

and differences of squares] [e.g., 2x2 + 4x, 2x – 2y + ax – ay, [x2 – x – 6, 2a2 + 11a + 5, 4x2 – 25]], using a variety of tools (e.g., concretematerials, computer algebra systems, paper and pencil) and strategies (e.g.,patterning).

Mathematical Process Focus • Selecting Tools and Computational Strategies • Connecting • Representing

MATH BACKGROUND | LESSON OVERVIEW

• This lesson focuses on determining common factors of algebraic terms ovenew topic in Grade 10. Students had experience determining common fact(GCF) of whole numbers in Grade 8, but not in Grade 9.

• In this lesson, students learn how to factor a polynomial by dividing out a several divisions may be necessary to determine the GFC and factor the pocommon factor of the terms is 1 or –1). All the polynomials in this lesson students will encounter polynomials that cannot be factored.

• Students should understand and be able to use the distributive property to are monomials or binomials.


GOAL Factor algebraic expressions by dividing out the greatest common factor.

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Student Book Pages 198–204


Pacing 5-10 min Introduction 30-40 min Teaching and Learning 15-20 min Consolidation

Materials algebra tiles

Recommended Practice Questions 7, 8, 9, 12, 15, 16, 18

Key Assessment Question Question 9

Extra Practice Lesson 4.1 Extra Practice

New Vocabulary/Symbols factor

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the set of integers, which is a rs and the greatest common factor

ommon factor. They learn that ynomial fully (until the only n be factored. In later lessons,

termine common factors that

1 Introducing the Lesson (5 to 10 min)

Introduce the concept of common factoring with the following context: • Ask students to imagine that a customer orders two rugs, one with an area

of 10 m2 and the other with an area of 15 m2. Ask: What are the possible dimensions with only whole numbers? You could have students sketch two different possibilities for each rug on grid paper.

• Discuss how students’ sketches are examples of factoring because the area is expressed as a product of two factors, the dimensions. Have students compare sketches and look for a pair of rugs with a common dimension (for example, 10 m by 1 m and 15 m by 1 m, or 5 m by 2 m and 5 m by 3 m). Then ask them to identify the greatest possible common dimension, or greatest common factor (for example, 5).

2 Teaching and Learning (30 to 40 min)

Learn About the Math Example 1 explores the answer to the lesson question: the sum of the squares of two consecutive integers is always divisible by 2. Pose the question, and have students individually choose two consecutive integers to test the conjecture. Then have students work in groups of three to brainstorm whether the conjecture is true and how they might convince someone that their answer is correct. To help students understand Abdul’s solution, demonstrate how the two groups are formed using concrete materials. The two groups are identical⎯both include x2, x, and 1⎯therefore the expression is divisible by 2. Have students discuss the Reflecting questions in their groups of three. Then discuss the solutions with the whole class.

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Answers to Reflecting A. Lisa used n and Abdul used x because the value of the first integer is

unknown. This allows the rule to apply to any integer that is substituted for the variable.

B. Lisa needed to show that 2 is a common factor for the terms in the expression in order to show that the expression is even, or divisible by 2.

C. Answers may vary, e.g., • I preferred Abdul’s strategy because he used algebra tiles, which made

the solution easy to see. • I preferred Lisa’s strategy because I prefer strategies that don’t require

me to use materials.

4.1: Common Factors in Polynomials | 133

3 Consolidation (15 to 20 min)

Apply the Math Using the Solved Examples

In Example 2, the GCF is represented using two different strategies: concretely (with algebra tiles) and algebraically. Have the students work in pairs, with one student using algebra tiles and the other student working through the solution algebraically. Partners should then explain to each other what they did. Discuss both strategies with the class, having different students explain their work. Invite students to talk about advantages of being shown different strategies. Ensure that students understand what is meant by factoring fully, as described in the second Communication Tip. For Examples 3 and 4, point out that the GCF can have more than one term. In Example 3, where the GCF is a monomial, have students check that the factoring is correct by expanding the answer. The product should match the original expression. Ask students to explain how the distributive property is used in the solution. In Example 4, the GCF is a binomial. To help students understand the grouping, suggest that they underline each group of terms with a coloured pencil in their notes. Ask students why the factor (x – 2) in part a) is in the answer once instead of twice, as it is in the original expression. To show that the original and factored expressions are equivalent, have half of the class expand 5x(x – 2) – 3(x – 2) and the other half expand (x – 2)(5x – 3). Then compare the results.

Answer to the Key Assessment Question The polynomials in parts a) and d) of question 9 have the same trinomial as one of their factors. Students’ answers may look slightly different if they use a negative sign. For example, in part b), the following answer is also correct: –5ac(2a – 4 + c2).

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9. a) dc2 – 2acd + 3a2d = d(c2 – 2ac + 3a2) b) –10a2c + 20ac – 5ac3 = 5ac(–2a + 4 – c2) c) 10ac2 –15a2c + 25 = 5(2ac2 – 3a2c + 5) d) 2a2c4 – 4a3c3 + 6a4c2 = 2a2c2(c2 – 2ac + 3a2) e) 3a5c3 – 2ac2 + 7ac = ac(3a4c2 – 2c + 7) f) 10c3d – 8cd2 + 2cd = 2cd(5c2 – 4d + 1)

Closing Have students read question 18 and discuss why expanding is the opposite of factoring. Emphasize that expanding involves multiplying and factoring involves dividing, as shown in the In Summary box, under Key Ideas. Ask students to suggest examples with numbers to show that multiplying is the opposite of dividing. Then have students demonstrate that expanding is the opposite of factoring by verifying some of the Practising questions they have completed.


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Assessment and Differentiating Instruction

What You Will See Students Doing …

When students understand…

Students determine the GCF of the terms in a polynomial.

Students factor polynomials by dividing out the GCF.

Students identify a GCF that has more than one term and factor appropriately.

If students misunderstand…

Students cannot determine the GCF in a polynomial, or they cannot distinguish the GCF from a factor that is not the GCF.

Students cannot factor polynomials by dividing out the GCF from each term. They might not divide each term by the same factor, or they might not divide correctly.

Students may not identify both terms in a GCF with more than one term, or they may not factor appropriately.

Key Assessment Question 9

Students identify the GCF of all terms in a polynomial.

Students factor a polynomial by dividing out the GCF from each term.

Students recognize when factors are identical.

Students are unsure of the GCF of all the terms in a polynomial, or they identify a factor that is not the GCF to divide out.

Students may miss dividing out the GCF from a term, or they might divide incorrectly.

Students may not realize that each term in one factor must match a term in another factor for the factors to be identical.

Differentiating Instruction | How You Can Respond

EXTRA SUPPORT 1. Remind students that they can check their factoring by using the distributive property to multiply the factors. 2. To help students understand how to determine the GCF, encourage them to look at the numerical coefficients of the terms in

the polynomial first, and then at the variable(s) to determine what all the terms have in common. 3. To help students factor a polynomial, explain that one of the factors is the GCF. To determine the other factor, they can

divide each term by the GCF. (When the GCF is a monomial, they will need to remember how to divide monomials.) 4. Provide algebra tiles whenever they might be helpful to give students a concrete model of factoring. Ask students to describe

their use of the algebra tiles and to relate the final result with the algebra tiles to the factored expression.

EXTRA CHALLENGE 1. If students excel at factoring, ask them to factor (4a2b – 1)2 – 8(4a2b – 1).

4.1: Common Factors in Polynomials | 135

4.2 EXPLORING THE FACTORIZATION OF TRINOMIALS Lesson at a Glance

Prerequisite Skills/Concepts • Identify the factors in an area model. • Represent a polynomial with degree 2 using algebra tiles. • Expand an algebraic expression using the distributive property.

Specific Expectation • Factor polynomial expressions involving [common factors,] trinomials,

[and differences of squares] [e.g., [2x2 + 4x, 2x – 2y + ax – ay], x2 – x – 6, 2a2 + 11a + 5, [4x2 – 25]], using a variety of tools (e.g., concrete materials,[computer algebra systems], paper and pencil) and strategies (e.g., patterning).



• In this lesson, students discover how to factor a trinomial using an algebrato arrange the algebra tiles for each trinomial to form a rectangle. Then thedimensions of the rectangle. (This factoring strategy using algebra tiles is strategy using algebra tiles that students learned in Chapter 3.).

• Next, students use a patterning strategy (looking for relationships) to deterto its binomial factors. This relationship will be covered more formally in

• In Reflecting, students encounter trinomials for which they cannot make ashould begin to realize that not all polynomials can be factored.

• Students should understand that answers can be checked by multiplying ththe product is the same as the original trinomial.


GOAL Discover the relationship between the coefficients and constants in a trinomial and the coefficients and constants in its factors.

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Pacing 5 min Introduction 35-45 min Teaching and Learning 10-20 min Consolidation

Material algebra tiles

Recommended Practice Questions 1, 2, 3, 4

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ile model. First, they explore how identify the factors as the e opposite of the expanding

ine how each trinomial is related esson 3. ectangle using algebra tiles. They

factors to determine whether

1 Introducing the Lesson (5 min)

Write the trinomial x2 + 4x + 3 on the board, and ask students why they cannot factor this trinomial by first determining the GCF. Explain that this trinomial can be factored another way, which they will discover using algebra tile models. • Distribute algebra tiles to pairs of students, and ask each pair to model the

trinomial. Then challenge students to rearrange their algebra tiles to form a rectangle. Have them sketch their rectangle model.

• Ask students to arrange the tiles in a different way to represent the same factors. Discuss why they can represent x + 3 along the side and x + 1 across. Some students might place the x2 tile at a different corner. Ensure that each student sees different ways to represent the trinomial with the same tiles.


Explore the Math Have students continue to work in pairs, discussing their observations and sharing their work. It is important for students to realize that a rectangle can be arranged in different ways and that the order of the factors does not matter. Guide students, as needed, when they start arranging the algebra tiles. Emphasize that when they are able to form a rectangle with the algebra tiles, the values along the sides show the factors. Ensure that they use different colours of tiles to represent positive and negative values. As students work, ask them to describe how all the tiles represent the expression and how the rectangles represent the factors. Discuss the style of writing the x term before the constant term in a factor (for example, x + 3, not 3 + x), but make sure that students realize that both represent the same value.

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Students may notice similarities in the trinomials in each part. In part A, each unit tile is positive. In part B, more tiles must be added to make the rectangle, and each unit tile is negative. Ensure that students add pairs of negative and positive tiles so the value of an expression does not change, as explained in the Communication Tip. Ask a few students to explain the Communication Tip in their own words. In part C, the coefficient of x2 is greater than 1, but students do not need to add pairs of tiles to create the rectangles. Encourage students to look for similarities as they compare trinomials. Ask them to consider how each trinomial is related to its binomial factors. In this exploration, students may be more comfortable using informal language to describe what they observe.

4.2: Exploring the Factorization of Trinomials | 137

Answers to Explore the Math A. i) (x + 1)(x + 1) iii) (x + 2)(x + 4) v) (x – 3)(x – 1)

ii) (x + 2)(x + 3) iv) (x – 1)(x – 1) vi) (x – 2)(x – 1)

B. i) (x – 3)(x + 1) iii) (x + 2)(x – 4) v) (x + 2)(x – 5)

ii) (x – 1)(x + 4) iv) (x + 3)(x – 2) vi) (x – 1)(x + 5)

C. i) (x + 1)(2x + 1) iii) (x – 1)(3x – 1) v) (x – 4)(2x + 1)

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ii) (x + 2)(2x + 1) iv) (x – 2)(2x – 3) vi) (x + 2)(3x – 1)


D. Answers may vary, e.g., i) The coefficient of x2 in each trinomial is equal to the product of the

coefficients of x in the factors. ii) The constant term in each trinomial is equal to the product of the

constant terms in the factors. iii) The coefficient of x in each trinomial is equal to the sum of these two

products: the coefficient in the first factor multiplied by the constant term in the second factor, and the coefficient in the second factor multiplied by the constant term in the first factor.

Answers to Reflecting E. Both factors are binomials. F. The coefficients in the two binomial factors have the product a. The

constants in the binomial factors have the product c (the constant term in the trinomial), but there may be several possibilities. To get the correct combination of constants in the binomials, systematically try different possibilities to determine which factors work. The product of the “inside” pair of numbers and the product of the “outside” pair of numbers must have the sum b.

G. A rectangular arrangement of algebra tiles cannot be created for x2 + 3x + 1 or 2x2 + x + 1. This implies that these trinomials cannot be factored.


• Remind students that since expanding is the opposite of factoring, expanding is a useful strategy for checking their answers.

• Students should answer the questions in Further Your Understanding independently.

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4.2: Exploring the Factorization of Trinomials | 139

4.3 FACTORING QUADRATICS: x2 + bx + c Lesson at a Glance

Prerequisite Skills/Concepts • Expand an algebraic expression using the distributive property. • Represent a polynomial with degree 2 using algebra tiles. • Identify the greatest common factor for a polynomial. • Sketch the parabola of a quadratic relation using its key properties

(x-intercepts, axis of symmetry, vertex).

Specific Expectation • Factor polynomial expressions involving common factors, trinomials, [and

differences of squares] [e.g., 2x2 + 4x, 2x – 2y + ax – ay, x2 – x – 6, 2a2 + 11a + 5, [4x2 – 25]], using a variety of tools (e.g., concrete materials,computer algebra systems, paper and pencil) and strategies (e.g., patterning).



• This lesson focuses on strategies for factoring trinomials of the form x2 + b• Students’ previous explorations with algebra tile models are further develo

to determine how each trinomial is related to its binomial factors. • It is important for students to understand that their answers can always be

to see if the product is the same as the original polynomial. Expanding is t• Students will encounter some trinomials that have values of a that are grea

of trinomial, they should look for a common factor among all the terms usLesson 4.1. When the GCF is factored out, the remaining trinomial has thecan continue factoring using the skills they develop in this lesson. This coworks when the first term in the trinomial is –x2.


GOAL Factor quadratic expressions of the form ax2 + bx + c, where a = 1.

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Recommended Practice Questions 4, 5, 7, 9, 10, 12, 13a, d



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+ c. ed as students look for patterns

hecked by multiplying the factors e opposite of factoring. r than 1. To factor this type g the skills they learned in

form x2 + bx + c. Students mon factor method also


Begin by briefly reviewing multiplication of binomials. • Tell students that the dimensions of a rectangle are (x + 1) and (x – 5).

Have each student determine the area of the rectangle by expanding: (x + 1)(x – 5) = x2 – 4x – 5.

• Then have students rewrite the area equation in the opposite order: x2 – 4x – 5 = (x + 1)(x – 5). Ask students to explain how factoring is the opposite of expanding. Have students model this with algebra tiles.


Investigate the Math Draw the area diagram, shown at the top of page 207 in the Student Book, on the board so that students can refer to it as they work through the lesson. Distribute algebra tiles to pairs of students. Have them work together, discussing their observations and sharing their work.

Answers to Investigate the Math A. (x + 3)(x + 4) (x + 3)(x + 5) (x + 3)(x + 6) (x + 4)(x + 4) (x + 4)(x + 5)

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Length Width Area: x2 + bx + c Value of b Value of c

x + 3 x + 4 x2 + 7x + 12 7 12

x + 3 x + 5 x2 + 8x + 15 8 15

x + 3 x + 6 x2 + 9x + 18 9 18

x + 4 x + 4 x2 + 8x + 16 8 16

x + 4 x + 5 x2 + 9x + 20 9 20

B. The value of b is the sum of the constant terms in the two factors. The value of c is the product of the constant terms in the two factors. i) (x + 6) and (x + 2) ii) (x + 3) and (x + 7) iii) (x + 5) and (x + 6) iv) (x + 9) and (x + 2)

4.3: Factoring Quadratics: x2 + bx + c | 141

C. x2 + x – 12 D. (x – 3)(x + 5) (x + 3)(x – 6) (x – 2)(x – 2) (x – 1)(x – 5)

Length Width Area: x2 + bx + c Value of b Value of c

x – 3 x + 5 x2 + 2x – 15 2 –15

x + 3 x – 6 x2 – 3x – 18 –3 –18

x – 2 x – 2 x2 – 4x + 4 –4 4

x – 1 x – 5 x2 – 6x + 5 –6 5

E. The value of b is the sum of the constant terms in the two factors. The value of c is the product of the constant terms in the two factors. i) (x – 5) and (x + 3) ii) (x + 6) and (x – 4) iii) (x – 6) and (x + 5) iv) (x – 7) and (x – 1)

F. For example, write two binomials that have x as the first term in each pair of parentheses. Write the second term in each binomial by determining two numbers that are factors of c and have the sum b.

Answers to Reflecting

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G. The signs can be used as follows: • If c in the area expression is positive, then the signs in the dimensions

will be either both positive or both negative. If b is positive, both signs will be positive. If b is negative, both signs will be negative.

• If c is negative, then the signs in the dimensions will be different; one will be positive, and the other will be negative. If b is positive, then the greater number will be positive. If b is negative, then the greater number will be negative.

H. No. For example, in a trinomial, b and c can have any integer values. There may not always be factors of c with the sum b, so it may not always be possible to factor a trinomial as the product of two binomials.




For Example 1, ask each student to factor the trinomial using algebra tiles. Have students check their factoring by expanding. In Examples 2 and 3, a systematic strategy is used to factor trinomials algebraically. Factors of the constant term in the trinomial are considered to determine which pair, if any, has the coefficient of x as its sum. Work through Example 2 with the class. Ask if there are any other pairs of numbers that could be the constant terms in the factors. For Example 3, have students work in pairs, with each partner completing either part a) or part b). When finished, the partners should exchange and check each other’s work by expanding. If an incorrect answer is noticed, the partners should discuss the correction together. Have the partners work through part c) together, checking all possibilities to determine that no factors are possible. Example 4 demonstrates factoring out the GCF before the next step in factoring the trinomial. If a ≠ 1, students should look to see if there is a GCF that can be factored out. Remind students to include the GCF as part of their answer.

Answer to the Key Assessment Question For question 12, students need to be aware that some expressions will require dividing out the greatest common factor. 12. a) a2 + 8a + 15 = (a + 3)(a + 5)

b) 3x2 – 21x – 54 = 3(x2 – 7x – 18) = 3(x – 9)(x + 2)

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c) z2 – 16z + 55 = (z – 11)(z – 5) d) x2 + 5x – 50 = (x – 5)(x + 10) e) x3 – 3x2 – 10x = x(x2 – 3x – 10)

= x(x – 5)(x + 2) f) 2xy2 – 26xy + 84x = 2x(y2 – 13y + 42)

= 2x(y – 6)(y – 7)

Closing Have pairs of students work together to complete question 18. For support, encourage them to look at the area diagram shown at the top of page 207 in the Student Book. Discuss how this systematic strategy using algebra can help them determine if a trinomial can be factored. Point out that even though all the trinomials in the Practising questions could be factored, students may encounter trinomials that cannot be factored, as they did in Example 3, part c).

4.3: Factoring Quadratics: x2 + bx + c | 143

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Students use algebra tiles to factor quadratic expressions in the form ax2 + bx + c, when a = 1.

Students factor quadratic expressions in the form ax2 + bx + c algebraically, when a = 1.

Students factor quadratic expressions in the form ax2 + bx + c algebraically, when a = 1 is a common factor, by factoring the GCF first.


Students cannot create rectangles with algebra tiles so that each side of the rectangle represents one factor. They might not be able to add a positive and negative pair of tiles when needed.

Students cannot algebraically determine the constant terms with the product c and the sum a for quadratic expressions in the form ax2 + bx + c, when a = 1.

Students cannot identify the GCF of the terms, or they forget to look for the GCF.


Students factor each trinomial. In parts b), e), and f), students factor the GCF and then factor the trinomial, which they recognize as having the form x2 + bx + c.

Students cannot factor all the expressions. In parts b), e), and f), students forget to factor the GCF first and then the trinomial, or they cannot identify the GCF. They may not recognize a common factor that has a variable.


EXTRA SUPPORT 1. Students can continue to use algebra tiles as a concrete strategy to identify factors or to check their work. 2. If students are not sure about the signs of the terms, have them create a list or chart of all the possibilities in their notes for

reference. For example: • If the factors have a product with a (+) sign and a sum with a (+) sign, then the signs of the constant terms in the factors

are (+)(+). • If the factors have a product with a (+) sign and a sum with a (–) sign, then the signs of the constant terms in the factors

are (–)(–). • If the factors have a product with a (–) sign and a sum with a (–) sign, then the signs of the constant terms in the factors

are (+)(–) or (–)(+) and the negative number must be greater in magnitude. • If the factors have a product with a (–) sign and a sum with a (+) sign, then the signs of the constant terms in the factors

are (+)(–) or (–)(+) and the positive number must be greater in magnitude. 3. Remind students that they can check their factoring by expanding, which means multiplying the factors.

EXTRA CHALLENGE 1. Challenge students to create trinomials of the form x2 + bx + c that cannot be factored, and to explain what they did. 2. Ask students to explain what happens if the x2 term is negative, as in the trinomial in question 13, part d). (Factor out the

GCF of –1 to rewrite –x2 – 9x – 14 as –(x2 + 9x + 14).)


MID-CHAPTER REVIEW

Big Ideas Covered So Far

• Factoring is the opposite of expanding. Expanding involves multiplying, and factoring involves determining the values to multiply.

• One strategy that can be used to factor an algebraic expression is to determine the greatest common factor of the terms in the expression. For example, 5x2 + 10x – 15 can be factored as 5(x2 + 2x – 3), since 5 is the greatest common factor of the terms.



Using the Frequently Asked Questions

Have students keep their Student Books closed. Display the Frequently Asked Questions on a board. Have students discuss the questions and use the discussion to draw out what the class thinks are good answers. Then have students compare the class answers with the answers on Student Book pages 214 and 215. Students can refer to the answers to the Frequently Asked Questions as they work through the Practice Questions.

Using the Mid-Chapter Review

Ask students if they have any questions about any of the topics covered so far in the chapter. Review any topics that students would benefit from considering again. Assign Practice Questions for class work and for homework.

To gain greater insight into students’ understanding of the material covered so far in the chapter, you may want to ask them questions such as the following:

• How is expanding related to multiplying? How is factoring related to dividing?

• How is multiplying related to dividing? How is factoring related to expanding?

• Is it possible to write the factors of a polynomial in more than one way? Explain.

• Is it possible to factor all quadratic expressions? How can you convince someone that you are correct?

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Chapter 4 Mid-Chapter Review | 145

4.4 FACTORING QUADRATICS: ax2 + bx + c Lesson at a Glance

Prerequisite Skills/Concepts • Understand and use the distributive property. • Identify the factors in an area model. • Identify a common factor in a polynomial. • Factor trinomials of the form x2 + bx + c.

Specific Expectation • Factor polynomial expressions involving common factors, trinomials, [and

differences of squares] [e.g., 2x2 + 4x, 2x – 2y + ax – ay, x2 – x – 6, 2a2 + 11a + 5, [4x2 – 25]], using a variety of tools (e.g., concrete materials,computer algebra systems, paper and pencil) and strategies (e.g., patterning).



• Students should be familiar with strategies for factoring quadratic expressiwhich were developed in Lesson 4.3. In this lesson, students build on theirexpressions of the form ax2 + bx + c, where a ≠ 1.

• To factor some trinomials, students will use the skills they developed in Lbefore they continue to factor the polynomial.

• In this lesson, the strategies for factoring are using algebra tiles, using an aalgebraic approach, and using decomposition.

• Students can check whether a quadratic expression ax2 + bx + c is factorabwhose product is ac and whose sum is b. If no numbers satisfy these condiexpression cannot be factored.


GOAL Factor quadratic expressions of the form ax2 + bx + c, where a ≠ 1.

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Recommended Practice Questions 6, 7, 9, 12, 13, 14



New Vocabulary/Symbols decompose

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ns of the form x2 + bx + c, kills as they factor quadratic

son 4.1 to divide out the GCF

ea model, using a systematic

by identifying two numbers ions, then the quadratic


Review the process for multiplying two binomials by asking the class to consider the example (3x + 2)(x – 3). • Guide students as they work together to multiply the factors using the

distributive property. • After writing the first expression, 3x2 – 9x + 2x – 6, ask students how to

simplify it by combining the two middle terms. • After writing the second expression, 3x2 – 7x – 6, point out that this is a

trinomial of the form ax2 + bx + c, where a = 3. Ask students to consider why the strategies they have previously learned for factoring will not work for this trinomial.

Remind students that factoring is the reverse of expanding. Tell students that they will learn a strategy in which they reverse the multiplying process to determine whether a quadratic expression with a ≠ 1, such as 3x2 – 7x – 6, can or cannot be factored.


Learn About the Math Discuss how Kellie graphed y = 3x2 + 11x + 6 using the skills learned in Chapter 3. Point out the following connection: If factoring yields y = a(x – )(x – ), then the x-intercepts can be determined using algebra. What is the value of x when y = 0? Example 1 shows how to factor the quadratic expression that was represented by the graph, 3x2 + 11x + 6, using three different strategies. • Have pairs of students use algebra tiles to work through Ellen’s solution.

One student could model this strategy electronically, while the other uses tiles.

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• Have the same pairs work through Neil’s solution using the systematic strategy they learned in the previous lesson, but with an area diagram. Make sure that students begin by choosing the correct first and last terms and then check to determine that the middle term works.

• With the whole class, work through Astrid’s solution. Discuss Astrid’s goal when she decomposed the coefficient of the x term: to create an expression with four terms that could be factored by grouping. Also discuss how decomposing the coefficient of the x term reverses the process of multiplying using the distributive property.

Have students continue to work in pairs to answer the Reflecting questions, in which they compare the three strategies. Remind students that they can check whether their factoring is correct by expanding the binomials.

4.4: Factoring Quadratics: ax2 + bx + c | 147

Answers to Reflecting A. Ellen selected tiles to represent 3x2 + 11x + 6. Then she arranged the tiles

to form a rectangle. This allowed her to determine a length and width whose product is equal to the quadratic expression, or the area of the rectangle.

B. Neil’s strategy is similar to the strategy used to factor trinomials in the previous lesson because he determined what integers can be multiplied to give the correct first and last terms and whose outer and inner terms have the middle term as the sum. Neil’s strategy is different because the coefficient of x2 is not 1, so he had to determine what combinations work for the first term in each binomial.

C. Astrid would need to determine pairs of factors of the product of (24)(3), which is 72. These pairs of factors are 1 and 72, 2 and 36, 3 and 24, 4 and 18, 6 and 12, and 8 and 9. Then Astrid would need to decide which of these pairs has a sum that is the coefficient of the middle term, 22. This pair is 4 and 18, so the trinomial 3x2 + 22x + 24 can be rewritten as 3x2 + 4x + 18x + 24. There are common factors, so the expression can be written as x(3x + 4) + 6(3x + 4). Grouping can be done to get (3x + 4)(x + 6).

D. Answers will vary, e.g., I like the decomposition strategy because I can determine which pair of factors of the product of the first and last numbers has a sum that is equal to the number in the middle term. I can tell for sure whether or not a trinomial can be factored. Also, this strategy is faster than using algebra tiles.



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Have students work through Examples 2, 3, and 4 in pairs. Then discuss each example as a class. Example 2 shows factoring using systematic trial. Remind students to choose first and last terms that are correct and then check to determine whether these terms give the correct middle term. Ask students how they know if they have chosen the correct terms. Example 3 shows factoring by decomposition. Students must understand factoring by grouping to use decomposition. Remind students that they are looking for integer factors of ac whose sum is b. Ask students to explain why organization is important when they are using this strategy (to keep track of the terms they are considering). Suggest that students underline each group of terms with a coloured pencil.


Example 4 shows factoring using an area diagram. This strategy is quicker to use than algebra tiles because the individual tiles do not need to be identified. Visual learners will benefit from using this strategy. An area diagram may also be helpful when using guess-and-test to determine the dimensions of a rectangle.

Answer to the Key Assessment Question Students can check factors for question 12 by expanding. 12. a) no, because 5 is not a factor of (1)(–52) = –52;

k2 + 9k – 52 = (k + 13)(k – 4) b) yes, 4k3 + 32k2 + 60k = 4k(k2 + 8k + 15) = 4k(k + 5)(k + 3) c) no, because 5 is not a factor of (6)(7) = 42;

6k2 + 23k + 7 = (2k + 7)(3k + 1) d) no, 10 + 19k – 15k2 = –15k2 + 19k + 10

= –(15k2 – 19k – 10) = –(3k – 5)(5k + 2) Positive factors of 150: 1 and 150, 2 and 75, 3 and 50, 5 and 30, 6 and 25, 10 and 15 One number must be negative since the product is negative. None of these pairs with one negative number have a sum of 19.

e) yes, 7k2 + 29k – 30 = 7k2 + 35k – 6k – 30 = 7k(k + 5) – 6(k + 5) = (7k – 6)(k + 5)

f) yes, 10k2 + 65k + 75 = 5(2k2 + 13k + 15) = 5(2k2 + 10k + 3k + 15) = 5[2k(k + 5) + 3(k + 5)] = 5(2k + 3)(k + 5)

Closing Question 16 provides an opportunity for students to think about the different factoring strategies they have learned. They may want to include examples of trinomials they have factored in this lesson, as well as a trinomial that cannot be factored. After students have finished their flow charts, have them present and discuss their ideas. Discuss how different flow charts can show the same information.

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4.4: Factoring Quadratics: ax2 + bx + c | 149

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Students use algebra tiles or an area diagram to create a rectangle for a given trinomial.

Students use a systematic approach to determine the correct factors of a quadratic expression of the form ax2 + bx + c, where a ≠ 1.

Students determine integer factors of ac, whose sum is b. Students then use these factors to decompose the expression and factor by grouping.


Students cannot connect algebra tiles to an area diagram, or they cannot complete the terms in a rectangle so that it matches the original expression.

Students may use a systematic approach to determine the correct first and last terms, but they may not be able to obtain the correct middle term.

Students may be able to determine the integer factors of ac, but they may not be able to figure out which two factors have the sum b. Students may not know how to use the integers whose product is ac and whose sum is b to determine the factors. Students may not know how to factor by grouping.


Students factor the polynomials and determine whether (k + 5) is one of the factors.

Alternatively, students use (k + 5) as one of the factors and try to determine another factor to multiply it by to get the original trinomial. If this is not possible, students can explain why.

Students may be unable to factor the polynomials.

Students may not think of trying to multiply (k + 5) by another factor to get the original trinomial.


EXTRA SUPPORT 1. Review multiplying binomials using the distributive property. Have students discuss how expanding and factoring are inverse

processes. 2. Review how to determine the factors of a number. Guide students as they list all the pairs of factors for several numbers. 3. Remind students to check whether their factoring is correct by multiplying the factors.

EXTRA CHALLENGE 1. Have students write quadratic expressions, determine whether the expressions can be factored, and then factor those that

can be factored.


4.5 FACTORING QUADRATICS: SPECIAL CASES Lesson at a Glance

Prerequisite Skills/Concepts • Understand and use the distributive property. • Represent a polynomial with degree 2 using algebra tiles. • Identify a common factor in a polynomial. • Factor trinomials of the form ax2 + bx + c, a ≠ 1.

Specific Expectation Factor polynomial expressions involving common factors, trinomials, and differences of squares [e.g., [2x2 + 4x, 2x – 2y + ax – ay], x2 – x – 6, 2a2 + 11a + 5, 4x2 – 25], using a variety of tools (e.g., concrete materials, computer algebra systems, paper and pencil) and strategies (e.g., patterning).


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• This lesson focuses on factoring perfect-square trinomials and differences patterns, students learn how to recognize these special cases.

• Students should already be familiar with strategies for factoring quadratic ax2 + bx + c, where a ≠ 1, which were developed in Lesson 4.4.

• In this lesson, students apply three strategies to factor perfect-square trinomforming a square with algebra tiles, using decomposition, and using logica

• As in the previous lessons, students should understand that factoring can amultiplying the factors to determine whether the result matches the origina

4.5: Fac

GOAL Factor perfect-square trinomials and differences of squares.

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Recommended Practice Questions 5 c), d), 6 c), d), 7, 8, 9, 10, 13, 15



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f squares. By discovering

pressions of the form

ials and differences of squares: reasoning. ays be checked by

expression.

ring Quadratics: Special Cases | 151


Ask students to factor x2 + 10x + 25 using the decomposition strategy they learned in the previous lesson. Guide them as they discuss the following steps: • List the positive factors of 25: 1 and 25, 5 and 5 • Determine the factors that add to 10: 5 and 5 • Rewrite the expression: x2 + 5x + 5x + 25 • Factor the expression: x(x + 5) + 5(x + 5) = (x + 5)(x + 5) Discuss the solution with the class. Point out that the factors are the same, so the expression can be written as (x + 5)2. Tell students that the original quadratic expression is a special case called a perfect-square trinomial. Have students generate other perfect-square trinomials by expanding other binomials, such as (x + 2)2 and (x – 3)2.


Learn About the Math Work through Parma’s solution in Example 1 with the class to ensure that students have a clear understanding of her strategy: test by substituting random numbers and then generalize using algebra tiles. Discuss Jarrod’s strategy with the students: look for patterns in the numbers and make a prediction. Have students expand the binomial to determine whether the result is the same as the original trinomial.

Answers to Reflecting A. You can write 4x2 + 12x + 9 as a square of the binomial 2x + 3, with no

remainder. It is a perfect square, just like 9 = 32 and 4 = 22. Both the length and the width of the algebra tiles represent 2x + 3, forming a square.

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B. The square root of a perfect square is one of the two terms in the pair of identical factors. Jarrod used the square root of 4, which is 2, and the square root of 9, which is 3, in the factors.

C. Both binomial factors are identical in a perfect square. When you use the distributive property, you multiply the constant in the binomial by the x term twice. This makes the coefficient of x even, because it is 2 times a number.


3 Consolidation (20-30 min)


Example 2 presents a trinomial that contains a negative value. Have students work in pairs to factor the trinomial. Suggest that they underline each group of terms with a coloured pencil to keep track of the terms. Students should notice that the resulting binomials are identical, so the original expression is also a perfect-square trinomial. Ask the class: Can the value of c in a perfect-square trinomial be negative? Why not? Example 3 shows the strategy for factoring a difference of squares using algebra tiles. Ask the class why the name “difference of squares” is appropriate. Then have pairs of students show the factoring using the tiles. Visual and hands-on learners should benefit from this approach. For Example 4, have pairs of students work through part a) and then part b). Ask the class: Why is there no middle term in a difference of squares? Discuss how the pattern in part a) helps students solve the more complex binomial in part b). Encourage students to describe the strategy in words: • Determine the square root of each term. • Write the square roots in the binomials. • Write opposite signs between the square roots, so the middle term equals 0. • Expand to check that the factoring is correct.

Answer to the Key Assessment Question If students have difficulty determining whether expressions in question 10 are perfect squares or differences of squares, they can refer to Example 2 and Example 4. 10. a) x4 – 12x2 + 36 = (x2 – 6)2

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b) a4 – 16 = (a2 – 22)(a2 + 4) = (a – 2)(a + 2)(a2 + 4) c) 49x2 – 100 = (7x – 10)(7x + 10) d) 12x2 – 60x2 + 75 = 3(4x2 – 20x + 25) = 3(2x – 5)2 e) x4 – 24x2 + 144 = (x2 – 12)2 f) 289x6 – 81 = (17x3 – 9)(17x3 + 9)

Closing Have students copy and complete question 15 individually. When they are finished, ask them to share their results with the class. Encourage students to check that the examples and non-examples are appropriate. This discussion will be revisited in the introduction to Lesson 4.6.

4.5: Factoring Quadratics: Special Cases | 153

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Curious Math

This Curious Math feature provides students with an opportunity to test a conjecture about composite numbers, first by calculating with different numbers and then by using algebra. Encourage students to use a calculator to test the conjecture.

Answers to Curious Math 1. Answers may vary, e.g., if 11 is the starting number, the result is 14 645. Since 14 645 = 5 × 2929, it is a composite

number. If 13 is the starting number, the result is 28 565. Since 28 565 = 5 × 5713, it is also a composite number. 2. Yes. A composite number is a number that has more than two factors. If you start with an even number that is greater

than 2, it is already a composite number. If you start with a prime number, such as 11, 13, or 101, the result ends in 5 so it is divisible by 5. The result is a composite number because 5 is a factor.

3. n4 + 4, where n > 1 4. n4 + 4n2 + 4 – 4n2 5. (n4 + 4n2 + 4) – 4n2 = (n2 + 2)2 – 4n2

I can factor further if I use the difference of squares pattern. a2 – b2 = (a + b)(a – b), since a = n2 + 2 and b = 2n So (n2 + 2 + 2n)(n2 + 2 – 2n) = (n2 + 2n + 2)(n2 – 2n + 2)

6. Since n4 + 4 = (n2 + 2n + 2)(n2 – 2n + 2) and n is a natural number greater than 1, each factor always produces a natural number greater than 1. This means that n4 + 4 will always have two factors, with neither factor equal to 1. So n4 + 4 will always be composite for all n > 1.


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Students use algebra tiles to create a square that represents the quadratic expression for a perfect-square trinomial.

Students use logical reasoning to determine the correct factors of a perfect-square trinomial or a difference of squares.

Students recognize a perfect-square trinomial or a difference of squares, and they know the pattern for factoring these special cases.


Students cannot use algebra tiles to create a square that represents the expression. They may not be able to create a square, or the square may not represent the expression.

Students can only use a guess-and-check strategy to determine the correct first and last terms.

Students may not recognize the special cases. Students may not remember how to factor perfect-square trinomials or differences of squares.


Students recognize a perfect-square trinomial in parts a), d), and e), and they factor it correctly.

Students recognize a difference of squares in parts b), c), and f), and they factor it correctly.

Students realize that they need to factor out the GCF before factoring the new trinomial in part d).

Students cannot recognize a perfect-square trinomial, or they cannot determine its factors.

Students cannot recognize a difference of squares, or they cannot determine its factors. In part f), they may not be able to determine the square root of 289.

Students do not factor out the common factor 3 in part d), or they may factor incorrectly.


EXTRA SUPPORT 1. Review the strategies for multiplying binomials that result in perfect-square trinomials and differences of squares. Have

students discuss patterns in the factors of these special cases, using their own words. 2. Review the strategies for calculating the square root of a number. Have students determine the square roots of several

numbers. Encourage them to use a calculator to determine the square roots of numbers they cannot calculate mentally. 3. Students could use algebra tiles to represent the quadratic expressions and their factors, to help them determine the factors,

or to check their work. 4. Remind students that they can multiply the factors to check whether their factoring is correct.

EXTRA CHALLENGE 1. Have students determine whether the polynomials 8x2 – 18x + 9 and 9x2 + 5x + 4 are perfect-square trinomials. (No, they

are not perfect-square trinomials because they cannot be written as a product of two identical factors. Alternatively, if you determine the product of the square roots of ac and then double it, and if the result is equal to b, then the quadratic is a perfect-square trinomial.)

2. Ask students how they can tell whether the polynomials 8x2 – 18x + 9 and 9x2 + 5x + 4 can be factored. (If there are two integer factors of ac whose sum is b, then the trinomial can be factored.)

3. Have students write expressions that are differences of squares and then factor them. Students could exchange work to check each other’s factoring.

4.5: Factoring Quadratics: Special Cases | 155

4.6 REASONING ABOUT FACTORING POLYNOMIALS Lesson at a Glance

Prerequisite Skills/Concepts • Understand and use the distributive property. • Identify a common factor in a polynomial. • Factor trinomials of the form ax2 + bx + c, a ≠ 1. • Factor perfect-square trinomials and differences of squares.

Specific Expectation • Factor polynomial expressions involving common factors, trinomials, and

differences of squares [e.g., 2x2 + 4x, 2x – 2y + ax – ay, x2 – x – 6, 2a2 + 11a + 5, 4x2 – 25], using a variety of tools (e.g., concrete materials, computer algebra systems, paper and pencil) and strategies (e.g., patterning).

Mathematical Process Focus • Reasoning and Proving • Selecting Tools and Computational Strategies • Connecting • Representing


• This lesson builds on the skills already developed in previous lessons in thfamiliar with strategies for factoring quadratic expressions of the form ax2 in Lesson 4.4). Also, students should be able to recognize an expression thdifference of squares (developed in Lesson 4.5).

• In this lesson, students use reasoning strategies to factor a variety of polynthat different strategies can be used to factor the same polynomial.

• This lesson focuses on the thinking that goes into deciding which factoringstrategy that students use to factor a polynomial will depend on the terms imay involve looking for common factors, using decomposition, or using thtrinomials and differences of squares.


GOAL Use reasoning to factor a variety of polynomials.

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Recommended Practice Questions 3, 4, 5, 6, 7, 11, 15



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hapter. Students should be x + c, where a ≠ 1 (developed

is a perfect-square trinomial or a

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rategies are appropriate. The he polynomial. The strategy atterns for perfect-square


Refer to the charts that students made for the closing question (question 15) in Lesson 4.5. Lead a class discussion about how to recognize a perfect-square trinomial and a difference of squares. Students should realize the following: • A perfect-square trinomial has a perfect square as its first and third terms.

If you multiply the square roots of these terms and then double the result, you get the middle term in the trinomial. The factors of a perfect-square trinomial contain the square roots of both numbers and have the same signs, so the binomials are identical. For example: 36x2 – 60x + 25 = (6x – 5)(6x – 5)

• A difference of squares has only two terms. Each term is a perfect square, and the terms are subtracted. The factors of a difference of squares contain the square roots of each term and the signs are different. For example: 9x2 – 16 = (3x – 4)(3x + 4)

Remind students that they can check their factoring by expanding.


Learn About the Math Discuss how the volume of a rectangular prism has three factors that represent length, width, and height. The given volume is a polynomial, but it is not written in the form ax2 + bx + c, where a ≠ 1. Ask students if they think that the polynomial can be rewritten in this form. Invite them to explain their reasoning. Have students work in pairs to solve the problem by factoring, comparing their solutions with Rob’s solution for Example 1.

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Answers to Reflecting A. Rob decided to use a factoring strategy because he needed to express the

trinomial for the volume as factors representing the length, width, and height of the prism.

B. Rob noticed that all the terms in the trinomial contained the variable x. C. Factors can be written in any order to give the same product:

volume = (length)(height)(width) or (width)(length)(height) or (height)(length)(width), and so on.

4.6: Reasoning about Factoring Polynomials | 157



Have students work in groups of three so that each student can complete one part of Example 2. When finished, they should pass their work to the student on their right for checking. If students notice incorrect answers, they should explain the reasons. Then students should pass the work to the right again for checking by expanding. Example 3 demonstrates how to factor a polynomial with more than one variable and shows that a polynomial can have more than two or three factors. Emphasize the need for being organized and writing every step clearly, using square brackets when necessary. Discuss the first step, where the GCF is divided out of every term. Then discuss how the polynomial within the parentheses can be further simplified by dividing out common factors from some of the terms. To help students keep track of what they are doing, suggest that they underline each group of terms using a coloured pencil. This visual cue will help them understand the grouping strategy.

Answer to the Key Assessment Question Question 7 involves applying factoring skills presented throughout the chapter. Students can use examples in Lesson 4.6, or in previous lessons, for reference. 7. a) 10x2 + 3x – 1 = 10x2 + 5x – 2x – 1

= 5x(2x + 1) – (2x + 1) = (5x – 1)(2x + 1)

b) 144a4 – 121 = (12a2 – 11)(12a2 + 11) c) 24ac – 8c + 21a – 7 = 8c(3a – 1) + 7(3a – 1)

= (8c + 7)(3a – 1)

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d) x3 – 11x2 + 18x = x(x2 – 11x + 18) = x(x – 9)(x – 2)

e) 18x2 + 60x + 50 = 2(9x2 + 30x + 25) = 2(3x + 5)2

f) x2y – 4y = y(x2 – 4) = y(x – 2)(x + 2)

Closing Have students complete question 15 on their own, and then discuss their results as a class. Ask students what they learned from seeing graphic organizers created by their classmates.


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Students factor polynomials by factoring the GCF first.

Students factor a polynomial of the form ax2 + bx + c algebraically using decomposition or logical reasoning.

Students can explain how to factor by grouping.

Students recognize a perfect-square trinomial or a difference of squares, and they know the pattern for factoring these special cases.


Students might not factor out the GCF first, or they might factor incorrectly.

Students cannot factor a polynomial of the form ax2 + bx + c using decomposition or logical reasoning.

Students may not know how to factor by grouping.

Students may not remember how to factor perfect-square trinomials or differences of squares, or they may not be able to identify these special cases.


In parts d), e), and f), students divide out the GCF.

In parts a), c), and d), students factor the polynomial using decomposition or logical reasoning.

Students recognize a perfect-square trinomial in part e) and a difference of squares in parts b) and f).

Students cannot determine the GCF of all the terms.

Students cannot factor polynomials using decomposition or logical reasoning.

Students cannot recognize a perfect-square trinomial or a difference of squares, or they do not follow a pattern to factor.


EXTRA SUPPORT 1. Review multiplying a monomial by a polynomial using the distributive property. Have students write the multiplication and

product in reverse order so they can see how the product is related to the monomial, which is the GCF in the factored form. 2. Review the process of decomposition for polynomials of the form ax2 + bx + c:

• Determine integer factors of ac whose sum is b. • Use these factors to decompose the middle term. • Factor by grouping.

3. Review multiplying binomials that result in perfect-square trinomials and differences of squares. Guide students as they discuss the pattern in the factors of the special cases.

4. Remind students that they can expand to determine whether their factoring is correct.

EXTRA CHALLENGE 1. Have students determine unknown factors by dividing a polynomial by a given factor. Ask them to determine each quotient,

assuming that the divisor is not zero. a) (x2 + x – 30) ÷ (x – 5) b) (36x2 – y2) ÷ (6x + y) c) (64x2 + 16x – 15) ÷ (8x – 3) d) (36x4 + 108x2 + 81) ÷ (6x2 + 9) Answers: a) (x + 6) b) (6x – y) c) (8x + 5) d) (6x2 + 9)

4.6: Reasoning about Factoring Polynomials | 159

CHAPTER REVIEW

Big Ideas Covered So Far

• Factoring is the opposite of expanding. Expanding involves multiplying, and factoring involves determining the values to multiply.

• One strategy that can be used to factor an algebraic expression is to determine the greatest common factor of the terms in the expression. For example, 5x2 + 10x – 15 can be factored as 5(x2 + 2x – 3), since 5 is the greatest common factor of the terms.



• If the quadratic expression ax2 + bx + c, where a ≠ 1, can be factored, then the factors have the form (px + r)(qx + s), where pq = a, rs = c, and ps + rq = b.

• A polynomial of the form a2 + 2ab + b2 or a2 – 2ab + b2 is a perfect-square trinomial. • a2 + 2ab + b2 can be factored as (a + b)2. • a2 – 2ab + b2 can be factored as (a – b)2.

• A polynomial of the form a2 – b2 is a difference of squares and can be factored as (a + b)(a – b).

Using the Frequently Asked Questions

Have students keep their Student Books closed. Display the Frequently Asked Questions on a board. Have students discuss the questions and use the discussion to draw out what the class thinks are good answers. Then have students compare the class answers with the answers on Student Book pages 238 and 239. Students can refer to the answers to the Frequently Asked Questions as they work through the Practice Questions.

Using the Chapter Review

Ask students if they have any questions about any of the topics covered so far in the chapter. Review any topics that students would benefit from considering again. Assign Practice Questions for class work and for homework.

To gain greater insight into students’ understanding of the material covered so far in the chapter, you may want to ask them questions such as the following:

• How can you quickly determine whether a polynomial is a difference of squares?

• How can you quickly determine whether a polynomial is a perfect-square trinomial?

• How can you always determine whether a polynomial is factored correctly?

• Is it possible to factor a polynomial like 4x2 – 100 without first dividing out the GCF from its terms? Explain.

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CHAPTER 4 TEST For further assessment items, please use Nelson’s Computerized Assessment Bank.

1. Determine the value of each symbol. a) x2 – x – 20 = (x + 2)(x – ) b) 8x2 + x + 15 = (4x + )( x + 5) c) 4x2 + x + 81 = ( x + )2 d) 9x2 – = ( x – 7)( x + )

2. The area of a rectangle is A = 14x2 – 7x. The length of one side is 7x. Determine the length of the other side.

3. Identify each trinomial that is modelled below, and state its factors. a) b)

4. Factor each expression. a) x2 + 2x – 15 b) 2(n – 7) + 3n(n – 7) c) y2 + 16y + 64 d) 2x2 – 9x – 5

5. a) Factor 4x2 + 24x + 36 using two different strategies. b) Which strategy do you prefer? Explain why.

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6. Factor each expression. a) 10x5 – 40x3 b) –5m2 + 41m – 8 c) 2x4 – 8x3 + 6x2 d) –15a3 + 3a

7. Factor, if possible. If factoring is not possible, explain why. a) x2 + 1 b) x2 + x – 18 c) 5x2 + x – 18

8. a) A parabola is defined by y = x2 + 4x – 21. Explain how you could determine the vertex of the parabola without using graphing technology.

b) State the vertex of the parabola in part a).

Chapter 4 Test | 161

C HAPTER 4 TEST ANSWERS 1. a) = 8, = 10

b) = 26, = 3, = 2 c) = 36, = 2, = 9 d) = 49, = 3, = 7

2. The length of the other side is 2x – 1.

3. a) 2x2 + x – 6 = (x + 2)(2x – 3) b) 6x2 – 5x + 1 = (2x – 1)(3x – 1)

4. a) (x + 5)(x – 3) b) (n – 7)(3n + 2) c) (y + 8)(y + 8) d) (2x + 1)(x – 5)

5. a) Answers may vary, e.g., Strategy 1: I determined two factors that worked and then divided out the common factors: 4x2 + 24x + 36 = (2x + 6)(2x + 6) = 2(x + 3)2(x + 3) = 4(x + 3)2 Strategy 2: I divided out the GCF first and then factored the perfect-square trinomial: 4x2 + 24x + 36 = 4(x2 + 6x + 9) = 4(x + 3)(x + 3) = 4(x + 3)2

b) Answers may vary, e.g., I prefer the second strategy because I only had to look for common factors once, and the trinomial was easier to factor.

6. a) 10x3(x2 – 4) = 10x3(x + 2)(x – 2) b) (–5m + 1)(m – 8) c) 2x2(x2 – 4x + 3) = 2x2(x – 3)(x – 1) d) 3a(–5a2 + 1)

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7. a) Factoring is not possible. Answers may vary, e.g., the factors of 1 are 1 and 1, or –1 and –1. Neither pair add to 0.

b) Factoring is not possible. Answers may vary, e.g., the factors of –18 are –1 and 18, –2 and 9, –3 and 6, and the pairs with opposite signs. None of these pairs have a sum that is equal to the coefficient of x, which is +1.

c) (5x – 9)(x + 2)

8. a) Answers may vary, e.g., factor the right side of the equation and determine the x-intercepts of the parabola from the factors. The vertex is on the line of symmetry. To get the x-coordinate of the vertex, add the x-intercepts and divide the sum by 2. Substitute this value into the original equation to determine the y-coordinate of the vertex.

b) The vertex is (–2, –25).


CHAPTER TASK

The Factoring Challenge

Student Book Page 243


Pacing 20-25 min Introducing the Chapter

Task 35-40 min Using the Chapter Task

Materials (for each player) 10 blank cards recording sheet calculator (optional)

Nelson Website http://www.nelson.com/math

Specific Expectation • Factor polynomial expressions involving common factors, trinomials, and

differences of squares [e.g., 2x2 + 4x, 2x – 2y + ax – ay, x2 – x – 6, 2a2 + 11a + 5, 4x2 – 25], using a variety of tools (e.g., concrete materials, paper and pencil) and strategies (e.g., patterning).

Introducing the Chapter Task (Whole Class/Individual) Introduce the Chapter Task on Student Book page 243. Explain to students that they will be playing a game called The Factoring Challenge and they will need to prepare some materials before they can begin the game. Have each student write 10 polynomials on cards (one per card), using the conditions given in Step 1. (Tell students that the information on their cards will be assessed, so they should write their name at the bottom of each card.) Then ask each student to prepare a recording sheet with their name on it, similar to the green recording sheet for Shirley, shown at the bottom of Student Book page 243. When students have finished preparing the materials, read through the instructions for the game together. Ask students to explain why Shirley earned 5 points for the factors she wrote on the recording sheet at the bottom of the page. They can use a calculator to determine this.

Using the Chapter Task (Small Groups) Tell students that they will be assessed on how well they play the game, so they will need to hand in their cards and recording sheets after they finish playing the game. Remind students to use the Task Checklist to help them produce an excellent result.

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Have students play the game in groups of three (or four, if necessary). Throughout the game, they can check and review each other’s work. As they are playing, observe them individually to see how they are interpreting and carrying out the task. You might want to record any polynomials that they find difficult to factor so you can discuss these polynomials as a class after the game.

Assessing Students’ Work Use the Assessment of Learning chart as a guide for assessing students’ work.

Chapter 4 Task | 163

Adapting the Task You can adapt the task in the Student Book to suit the needs of your students. For example: • Group students of different abilities so that stronger students can help other

students. • You might want to set a time limit on each turn. If students are having

difficulty factoring the polynomial on a particular card, they could draw another card.

• You might want to assess only 8 of the 10 turns and have students mark an X beside the two turns that were most problematic.

• Individual students could complete the recording sheet using their own set of 10 cards.

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Assessment of Learning—What to Look for in Students’ Work…

Assessment Strategy: Interview/Observation and Product Marking

Level of Performance

1 2 3 4

demonstrates limited knowledge of content

demonstrates some knowledge of content

demonstrates considerable knowledge of content

demonstrates thorough knowledge of content

Knowledge and Understanding Knowledge of content

Understanding of mathematical concepts

demonstrates limited understanding of concepts (e.g., is able to use a factoring strategy to factor only a few of the polynomials; is unable to determine when a polynomial is not factorable)

demonstrates some understanding of concepts (e.g., is able to use a factoring strategies to factor some of the polynomials; has difficulty determining when a polynomial is not factorable)

demonstrates considerable understanding of concepts (e.g., is able to use some factoring strategies to factor most of the polynomials; is generally able to determine when a polynomial is not factorable)

demonstrates thorough understanding of concepts (e.g., uses appropriate factoring strategies to factor all of the polynomials; is able to determine when a polynomial is not factorable)

uses planning skills with limited effectiveness

uses planning skills with some effectiveness

uses planning skills with considerable effectiveness

uses planning skills with a high degree of effectiveness

uses processing skills with limited effectiveness

uses processing skills with some effectiveness

uses processing skills with considerable effectiveness

uses processing skills with a high degree of effectiveness

Thinking Use of planning skills • understanding the

problem • making a plan for

solving the problem Use of processing skills • carrying out a plan • looking back at the

solution

Use of critical/creative thinking processes

uses critical/creative skills with limited effectiveness (e.g., does not work backwards using a factoring strategy to create polynomials that fit the conditions; uses strictly guess-and-check)

uses critical/creative skills with some effectiveness (e.g., works backwards using one factoring strategy to create some polynomials that fit the conditions, but still relies heavily on guess-and-check)

uses critical/creative skills with considerable effectiveness (e.g., works backwards using more than one factoring strategy, such as recognizing differences of squares and perfect-square trinomials)

uses critical/creative skills with a high degree of effectiveness (e.g., works backwards using several factoring strategies, such as recognizing differences of squares, perfect-square trinomials, and dividing out the GCF)

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Chapter 4 Assessment of Learning | 165

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Assessment of Learning—What to Look for in Students’ Work…

Assessment Strategy: Interview/Observation and Product Marking

Level of Performance

1 2 3 4

expresses and organizes mathematical thinking with limited effectiveness

expresses and organizes mathematical thinking with some effectiveness

expresses and organizes mathematical thinking with considerable effectiveness

expresses and organizes mathematical thinking with a high degree of effectiveness

communicates for different audiences and purposes with limited effectiveness (e.g., attempts to explain own work to the group; is not able to explain most of the group’s work)

communicates for different audiences and purposes with some effectiveness (e.g., is able to explain own work to the group; checks and reviews some of the group’s work)

communicates for different audiences and purposes with considerable effectiveness (e.g., is able to explain most of the group’s work)

communicates for different audiences and purposes with a high degree of effectiveness (e.g., is able to explain clearly all of the group’s work)

Communication Expression and organization of ideas and mathematical thinking, using oral, visual, and written forms

Communication for different audiences and purposes in oral, visual, and written forms

Use of conventions, vocabulary, and terminology of the discipline in oral, visual, and written forms

uses conventions, vocabulary, and terminology of the discipline with limited effectiveness (e.g., is able to record few of the polynomials and factors correctly; calculates the points earned with assistance)

uses conventions, vocabulary, and terminology of the discipline with some effectiveness (e.g., is able to record some of the polynomials, factors, and points earned correctly)

uses conventions, vocabulary, and terminology of the discipline with considerable effectiveness (e.g., correctly records most of the polynomials, factors, and points earned)

uses conventions, vocabulary, and terminology of the discipline with a high degree of effectiveness (e.g., correctly records all of the polynomials, factors, and points earned)

applies knowledge and skills in familiar contexts with limited effectiveness

applies knowledge and skills in familiar contexts with some effectiveness

applies knowledge and skills in familiar contexts with considerable effectiveness

applies knowledge and skills in familiar contexts with a high degree of effectiveness

transfers knowledge and skills to new contexts with limited effectiveness

transfers knowledge and skills to new contexts with some effectiveness

transfers knowledge and skills to new contexts with considerable effectiveness

transfers knowledge and skills to new contexts with a high degree of effectiveness

Application Application of knowledge and skills in familiar contexts

Transfer of knowledge and skills to new contexts

Making connections within and between various contexts

makes connections within and between various contexts with limited effectiveness

makes connections within and between various contexts with some effectiveness

makes connections within and between various contexts with considerable effectiveness

makes connections within and between various contexts with a high degree of effectiveness


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