# Chapter 7 Review Exercises

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## Chapter 7 Review Exercises

### 7.1 Greatest Common Factor and Factor by Grouping

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

42, 60

6

450, 420

90, 150, 105

15

##### Exercise 4

60, 294, 630

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

##### Exercise 5

$$24x−42$$

$$6(4x−7)$$

##### Exercise 6

$$35y+84$$

##### Exercise 7

$$15m^4+6m^{2}n$$

$$3m^2(5m2+2n)$$

##### Exercise 8

$$24pt^4+16t^7$$

Factor by Grouping

In the following exercises, factor by grouping.

##### Exercise 9

$$ax−ay+bx−by$$

$$(a+b)(x−y)$$

##### Exercise 10

$$x^{2}y−xy^2+2x−2y$$

##### Exercise 11

$$x^2+7x−3x−21$$

$$(x−3)(x+7)$$

##### Exercise 12

$$4x^2−16x+3x−12$$

##### Exercise 13

$$m^3+m^2+m+1$$

$$(m^2+1)(m+1)$$

##### Exercise 14

$$5x−5y−y+x$$

### 7.2 Factor Trinomials of the form $$x^2+bx+c$$

Factor Trinomials of the Form $$x^2+bx+c$$

In the following exercises, factor each trinomial of the form $$x^2+bx+c$$

##### Exercise 15

$$u^2+17u+72$$

$$(u+8)(u+9)$$

##### Exercise 16

$$a^2+14a+33$$

##### Exercise 17

$$k^2−16k+60$$

$$(k−6)(k−10)$$

##### Exercise 18

$$r^2−11r+28$$

##### Exercise 19

$$y^2+6y−7$$

$$(y+7)(y−1)$$

##### Exercise 20

$$m^2+3m−54$$

##### Exercise 21

$$s^2−2s−8$$

$$(s−4)(s+2)$$

##### Exercise 22

$$x^2−3x−10$$

Factor Trinomials of the Form $$x^2+bxy+cy^2$$

In the following examples, factor each trinomial of the form $$x^2+bxy+cy^2$$

##### Exercise 23

$$x^2+12xy+35y^2$$

$$(x+5y)(x+7y)$$

##### Exercise 24

$$u^2+14uv+48v^2$$

##### Exercise 25

$$a^2+4ab−21b^2$$

$$(a+7b)(a−3b)$$

##### Exercise 26

$$p^2−5pq−36q^2$$

### 7.3 Factoring Trinomials of the form $$ax^2+bx+c$$

Recognize a Preliminary Strategy to Factor Polynomials Completely

In the following exercises, identify the best method to use to factor each polynomial.

##### Exercise 27

$$y^2−17y+42$$

Undo FOIL

##### Exercise 28

$$12r^2+32r+5$$

##### Exercise 29

$$8a^3+72a$$

Factor the GCF

##### Exercise 30

$$4m−mn−3n+12$$

Factor Trinomials of the Form $$ax^2+bx+c$$ with a GCF

In the following exercises, factor completely.

##### Exercise 31

$$6x^2+42x+60$$

$$6(x+2)(x+5)$$

##### Exercise 32

$$8a^2+32a+24$$

##### Exercise 33

$$3n^4−12n^3−96n^2$$

$$3n^{2}(n−8)(n+4)$$

##### Exercise 34

$$5y^4+25y^2−70y$$

Factor Trinomials Using the “ac” Method

In the following exercises, factor.

##### Exercise 35

$$2x^2+9x+4$$

$$(x+4)(2x+1)$$

##### Exercise 36

$$3y^2+17y+10$$

##### Exercise 37

$$18a^2−9a+1$$

$$(3a−1)(6a−1)$$

##### Exercise 38

$$8u^2−14u+3$$

##### Exercise 39

$$15p^2+2p−8$$

$$(5p+4)(3p−2)$$

##### Exercise 40

$$15x^2+6x−2$$

##### Exercise 41

$$40s^2−s−6$$

$$(5s−2)(8s+3)$$

##### Exercise 42

$$20n^2−7n−3$$

Factor Trinomials with a GCF Using the “ac” Method

In the following exercises, factor.

##### Exercise 43

$$3x^2+3x−36$$

$$3(x+4)(x−3)$$

##### Exercise 44

$$4x^2+4x−8$$

##### Exercise 45

$$60y^2−85y−25$$

$$5(4y+1)(3y−5)$$

##### Exercise 46

$$18a^2−57a−21$$

### 7.4 Factoring Special Products

Factor Perfect Square Trinomials

In the following exercises, factor.

##### Exercise 47

$$25x^2+30x+9$$

$$(5x+3)^2$$

##### Exercise 48

$$16y^2+72y+81$$

##### Exercise 49

$$36a^2−84ab+49b^2$$

$$(6a−7b)^2$$

##### Exercise 50

$$64r^2−176rs+121s^2$$

##### Exercise 51

$$40x^2+360x+810$$

$$10(2x+9)^2$$

##### Exercise 52

$$75u^2+180u+108$$

##### Exercise 53

$$2y^3−16y^2+32y$$

$$2y(y−4)^2$$

##### Exercise 54

$$5k^3−70k^2+245k$$

Factor Differences of Squares

In the following exercises, factor.

##### Exercise 55

$$81r^2−25$$

$$(9r−5)(9r+5)$$

##### Exercise 56

$$49a^2−144$$

##### Exercise 57

$$169m^2−n^2$$

$$(13m+n)(13m−n)$$

##### Exercise 58

$$64x^2−y^2$$

##### Exercise 59

$$25p^2−1$$

$$(5p−1)(5p+1)$$

##### Exercise 60

$$1−16s^2$$

##### Exercise 61

$$9−121y^2$$

$$(3+11y)(3−11y)$$

##### Exercise 62

$$100k^2−81$$

##### Exercise 64

$$20x^2−125$$

$$5(2x−5)(2x+5)$$

##### Exercise 64

$$18y^2−98$$

##### Exercise 65

$$49u^3−9u$$

$$u(7u+3)(7u−3)$$

##### Exercise 66

$$169n^3−n$$

Factor Sums and Differences of Cubes

In the following exercises, factor.

##### Exercise 67

$$a^3−125$$

$$(a−5)(a^2+5a+25)$$

##### Exercise 68

$$b^3−216$$

##### Exercise 69

$$2m^3+54$$

$$2(m+3)(m^2−3m+9)$$

##### Exercise 70

$$81x^3+3$$

### 7.5 General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

##### Exercise 71

$$24x^3+44x^2$$

$$4x^{2}(6x+11)$$

##### Exercise 72

$$24a^4−9a^3$$

##### Exercise 73

$$16n^2−56mn+49m^2$$

$$(4n−7m)^2$$

##### Exercise 74

$$6a^2−25a−9$$

##### Exercise 75

$$5r^2+22r−48$$

(r+6)(5r−8)

##### Exercise 76

$$5u^4−45u^2$$

##### Exercise 77

$$n^4−81$$

$$(n^2+9)(n+3)(n−3)$$

##### Exercise 78

$$64j^2+225$$

##### Exercise 79

$$5x^2+5x−60$$

$$5(x−3)(x+4)$$

##### Exercise 80

$$b^3−64$$

##### Exercise 81

$$m^3+125$$

$$(m+5)(m^2−5m+25)$$

##### Exercise 82

$$2b^2−2bc+5cb−5c^2$$

Use the Zero Product Property

In the following exercises, solve.

##### Exercise 83

$$(a−3)(a+7)=0$$

$$a=3$$, $$a=−7$$

##### Exercise 84

$$(b−3)(b+10)=0$$

##### Exercise 85

$$3m(2m−5)(m+6)=0$$

$$m=0$$, $$m=−6$$, $$m=\frac{5}{2}$$

##### Exercise 86

$$7n(3n+8)(n−5)=0$$

In the following exercises, solve.

##### Exercise 87

$$x^2+9x+20=0$$

$$x=−4$$, $$x=−5$$

##### Exercise 88

$$y^2−y−72=0$$

##### Exercise 89

$$2p^2−11p=40$$

$$p=−\frac{5}{2}$$, p=8

##### Exercise 90

$$q^3+3q^2+2q=0$$

##### Exercise 91

$$144m^2−25=0$$

$$m=\frac{5}{12}$$, $$m=−\frac{5}{12}$$

##### Exercise 92

$$4n^2=36$$

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

##### Exercise 93

The product of two consecutive numbers is 462.

−21,−22

21, 22

##### Exercise 94

The area of a rectangular shaped patio 400 square feet. The length of the patio is 99 feet more than its width. Find the length and width.

## Practice Test

In the following exercises, find the Greatest Common Factor in each expression.

##### Exercise 95

$$14y−42$$

$$7(y−6)$$

##### Exercise 96

$$−6x^2−30x$$

##### Exercise 97

$$80a^2+120a^3$$

$$40a^{2}(2+3a)$$

##### Exercise 98

$$5m(m−1)+3(m−1)$$

In the following exercises, factor completely.

##### Exercise 99

$$x^2+13x+36$$

$$(x+7)(x+6)$$

##### Exercise 100

$$p^2+pq−12q^2$$

##### Exercise 101

$$3a^3−6a^2−72a$$

$$3a(a+4)(a-6)$$

##### Exercise 102

$$s^2−25s+84$$

##### Exercise 103

$$5n^2+30n+45$$

$$5(n+3)^2$$

##### Exercise 104

$$64y^2−49$$

##### Exercise 105

$$xy−8y+7x−56$$

$$(x−8)(y+7)$$

##### Exercise 106

$$40r^2+810$$

##### Exercise 107

$$9s^2−12s+4$$

$$(3s−2)^2$$

##### Exercise 1008

$$n^2+12n+36$$

##### Exercise 109

$$100−a^2$$

$$(10−a)(10+a)$$

##### Exercise 110

$$6x^2−11x−10$$

##### Exercise 111

$$3x^2−75y^2$$

$$3(x+5y)(x−5y)$$

##### Exercise 112

$$c^3−1000d^3$$

##### Exercise 113

$$ab−3b−2a+6$$

$$(a−3)(b−2)$$

##### Exercise 114

$$6u^2+3u−18$$

##### Exercise 115

$$8m^2+22m+5$$

$$(4m+1)(2m+5)$$

In the following exercises, solve.

##### Exercise 116

$$x^2+9x+20=0$$

##### Exercise 117

$$y^2=y+132$$

$$y=−11$$, $$y=12$$

##### Exercise 118

$$5a^2+26a=24$$

##### Exercise 119

$$9b^2−9=0$$

$$b=1$$, $$b=−1$$

##### Exercise 120

$$16−m^2=0$$

##### Exercise 121

$$4n^2+19n+21=0$$

$$n=−\frac{7}{4}$$, n=−3

##### Exercise 122

$$(x−3)(x+2)=6$$

##### Exercise 123

The product of two consecutive integers is 156.