5-4 Factoring Quadratic Expressions

M11.A.1.2.1: Find the Greatest Common Factor and/or the Least Common Multiple for sets of monomials

M11.D.2.1.5: Solve quadratic equations using factoring

M11.D.2.2.2: Factor algebraic expressions, including differences of squares and trinomials

Objectives

Finding Common Binomial Factors

Factoring Special Expressions

Vocabulary

Factoring is rewriting an expression as the product of its factors.

The greatest common factor (GCF) of an expression is a common factor of the terms of the expression.

Factor each expression.

a. 15x2 + 25x + 100

15x2 + 25x + 100 = 5(3x2) + 5(5x) + 5(20) Factor out the GCF, 5

= 5(3x2 + 5x + 20)Rewrite using the Distributive Property.

b. 8m2 + 4m

8m2 + 4m = 4m(2m) + 4m(1) Factor out the GCF, 4m

= 4m(2m + 1)Rewrite using the Distributive Property.

Finding Common Factors

Factor x2 + 10x + 24.

Step 2: Rewrite the term bx using the factors you found. Group the remaining terms and find the common factors for each group.

Step 1: Find factors with product ac and sum b.

Factors of 24

Sum of factors

1, 24

25

2, 12

14

3, 8

11

6, 4

10

Since ac = 24 and b = 10, find positive factors with product 24 and sum 11.

} }x2 + 10x + 24x2 + 4x + 6x + 24 Rewrite bx : 10x = 4x + 6x.

x(x + 4) + 6(x + 4) Find common factors.

Factoring when ac > 0 and b > 0

(continued)

Step 3: Rewrite the expression as a product of two binominals.

(x + 6)(x + 4) Rewrite using the Distributive Property.

x(x + 4) + 6(x + 4)

Check: (x + 6)(x + 4) = x2 + 4x + 6x + 24= x2 + 10x + 24

Continued

Factor x2 – 14x + 33.

Step 1: Find factors with product ac and sum b.

Step 2: Rewrite the term bx using the factors you found. Then find common factors and rewrite the expression as a product of two binomials.

Factors of 33

Sum of factors

–1, –33

–34

–3, –11

–14

Since ac = 33 and b = –14, find negative factors with product 33 and sum b.

} }x2 - 14x + 33x2 – 3x – 11x + 33Rewrite bx.

x(x – 3) – 11(x – 3) Find common factors.

(x – 11)(x – 3) Rewrite using the Distributive Property.

Factoring when ac > 0 and b < 0

Factor x2 + 3x –28.

Step 1: Find factors with product ac and sum b.

Step 2: Since a = 1, you can write binomials using the factors you found.

x2 + 3x – 28

(x – 4)(x + 7) Use the factors you found.

Factors of –28

Sum of factors

1, –28

–27

–1, 28

27

2, –14

–12

–2, 14

12

4, –7

–3

–4, 7

3

Since ac = –28 and b = 3, find factors 2 with product –28 and sum 3.

Factoring When ac < 0

Factor 6x2 – 31x + 35.

Step 1: Find factors with product ac and sum b.

Step 2: Rewrite the term bx using the factors you found. Then find common factors and rewrite the expression as the product of two binomials.

2x(3x – 5) – 7(3x – 5) Find common factors.

(2x – 7)(3x – 5) Rewrite using the Distributive Property.

6x2 – 31x + 35

6x2 – 10x – 21x + 35 Rewrite bx.} }

Factors of 210

Sum of factors

-1, –210

–211

–2, –105

–107

–3, –70

–73

–5, –42

–47

–10, –21

–31

Since ac = 210 and b = –31, find negative factors with product 210 and sum –31.

Factoring When a ≠ 1 and ac > 0

Factor 6x2 + 11x – 35.Step 1: Find factors with product ac and sum b.

Step 2: Rewrite the term bx using the factors you found. Then find common factors and rewrite the expression as the product of two binomials.

2x(3x – 5) + 7(3x – 5) Find common factors.(2x + 7)(3x – 5) Rewrite using the Distributive Property.

6x2 + 11x - 356x2 – 10x + 21x – 35 Rewrite bx.

Factors of –210

Sum of factors

1, –210

–209

–1, 210

209

2, –105

–103

–2, 105

103

3, –70

–67

Since ac = -210 and b = 11, find factors with product –210 and sum 11.

Factors of –210

Sum of factors

–3, 70

67

5, –42

–37

–5, 42

37

10, –21

–11

–10, 21

11

Factoring When a ≠ 1 and ac < 0

Vocabulary

A perfect square trinomial is the product you obtain when you square a binomial.

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

Factor 100x2 + 180x + 81.

100x2 + 180x + 81 = (10x)2 + 180 + (9)2 Rewrite the first and third terms as squares.

= (10x)2 + 180 + (9)2

Rewrite the middle term to verify the perfect square trinomial pattern.= (10x + 9)2 a2 + 2ab + b2 = (a + b)2

Factoring a Perfect Square Trinomial

Homework

Pg 259 #2-42 EOE