Loading [MathJax]/extensions/TeX/boldsymbol.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

Chapter 7 Review Exercises

( \newcommand{\kernel}{\mathrm{null}\,}\)

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.

Exercise 1

42, 60

Answer

6

Exercise 2

450, 420

Exercise 3

90, 150, 105

Answer

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

Answer

6(4x−7)

Exercise 6

35y+84

Exercise 7

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

Answer

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

Answer

(a+b)(x−y)

Exercise 10

x^{2}y−xy^2+2x−2y

Exercise 11

x^2+7x−3x−21

Answer

(x−3)(x+7)

Exercise 12

4x^2−16x+3x−12

Exercise 13

m^3+m^2+m+1

Answer

(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

Answer

(u+8)(u+9)

Exercise 16

a^2+14a+33

Exercise 17

k^2−16k+60

Answer

(k−6)(k−10)

Exercise 18

r^2−11r+28

Exercise 19

y^2+6y−7

Answer

(y+7)(y−1)

Exercise 20

m^2+3m−54

Exercise 21

s^2−2s−8

Answer

(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

Answer

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

Exercise 24

u^2+14uv+48v^2

Exercise 25

a^2+4ab−21b^2

Answer

(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

Answer

Undo FOIL

Exercise 28

12r^2+32r+5

Exercise 29

8a^3+72a

Answer

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

Answer

6(x+2)(x+5)

Exercise 32

8a^2+32a+24

Exercise 33

3n^4−12n^3−96n^2

Answer

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

Answer

(x+4)(2x+1)

Exercise 36

3y^2+17y+10

Exercise 37

18a^2−9a+1

Answer

(3a−1)(6a−1)

Exercise 38

8u^2−14u+3

Exercise 39

15p^2+2p−8

Answer

(5p+4)(3p−2)

Exercise 40

15x^2+6x−2

Exercise 41

40s^2−s−6

Answer

(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

Answer

3(x+4)(x−3)

Exercise 44

4x^2+4x−8

Exercise 45

60y^2−85y−25

Answer

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

Answer

(5x+3)^2

Exercise 48

16y^2+72y+81

Exercise 49

36a^2−84ab+49b^2

Answer

(6a−7b)^2

Exercise 50

64r^2−176rs+121s^2

Exercise 51

40x^2+360x+810

Answer

10(2x+9)^2

Exercise 52

75u^2+180u+108

Exercise 53

2y^3−16y^2+32y

Answer

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

Answer

(9r−5)(9r+5)

Exercise 56

49a^2−144

Exercise 57

169m^2−n^2

Answer

(13m+n)(13m−n)

Exercise 58

64x^2−y^2

Exercise 59

25p^2−1

Answer

(5p−1)(5p+1)

Exercise 60

1−16s^2

Exercise 61

9−121y^2

Answer

(3+11y)(3−11y)

Exercise 62

100k^2−81

Exercise 64

20x^2−125

Answer

5(2x−5)(2x+5)

Exercise 64

18y^2−98

Exercise 65

49u^3−9u

Answer

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

Answer

(a−5)(a^2+5a+25)

Exercise 68

b^3−216

Exercise 69

2m^3+54

Answer

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

Answer

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

Exercise 72

24a^4−9a^3

Exercise 73

16n^2−56mn+49m^2

Answer

(4n−7m)^2

Exercise 74

6a^2−25a−9

Exercise 75

5r^2+22r−48

Answer

(r+6)(5r−8)

Exercise 76

5u^4−45u^2

Exercise 77

n^4−81

Answer

(n^2+9)(n+3)(n−3)

Exercise 78

64j^2+225

Exercise 79

5x^2+5x−60

Answer

5(x−3)(x+4)

Exercise 80

b^3−64

Exercise 81

m^3+125

Answer

(m+5)(m^2−5m+25)

Exercise 82

2b^2−2bc+5cb−5c^2

7.6 Quadratic Equations

Use the Zero Product Property

In the following exercises, solve.

Exercise 83

(a−3)(a+7)=0

Answer

a=3, a=−7

Exercise 84

(b−3)(b+10)=0

Exercise 85

3m(2m−5)(m+6)=0

Answer

m=0, m=−6, m=\frac{5}{2}

Exercise 86

7n(3n+8)(n−5)=0

Solve Quadratic Equations by Factoring

In the following exercises, solve.

Exercise 87

x^2+9x+20=0

Answer

x=−4, x=−5

Exercise 88

y^2−y−72=0

Exercise 89

2p^2−11p=40

Answer

p=−\frac{5}{2}, p=8

Exercise 90

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

Exercise 91

144m^2−25=0

Answer

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.

Answer

−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

Answer

7(y−6)

Exercise 96

−6x^2−30x

Exercise 97

80a^2+120a^3

Answer

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

Exercise 98

5m(m−1)+3(m−1)

In the following exercises, factor completely.

Exercise 99

x^2+13x+36

Answer

(x+7)(x+6)

Exercise 100

p^2+pq−12q^2

Exercise 101

3a^3−6a^2−72a

Answer

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

Exercise 102

s^2−25s+84

Exercise 103

5n^2+30n+45

Answer

5(n+3)^2

Exercise 104

64y^2−49

Exercise 105

xy−8y+7x−56

Answer

(x−8)(y+7)

Exercise 106

40r^2+810

Exercise 107

9s^2−12s+4

Answer

(3s−2)^2

Exercise 1008

n^2+12n+36

Exercise 109

100−a^2

Answer

(10−a)(10+a)

Exercise 110

6x^2−11x−10

Exercise 111

3x^2−75y^2

Answer

3(x+5y)(x−5y)

Exercise 112

c^3−1000d^3

Exercise 113

ab−3b−2a+6

Answer

(a−3)(b−2)

Exercise 114

6u^2+3u−18

Exercise 115

8m^2+22m+5

Answer

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

In the following exercises, solve.

Exercise 116

x^2+9x+20=0

Exercise 117

y^2=y+132

Answer

y=−11, y=12

Exercise 118

5a^2+26a=24

Exercise 119

9b^2−9=0

Answer

b=1, b=−1

Exercise 120

16−m^2=0

Exercise 121

4n^2+19n+21=0

Answer

n=−\frac{7}{4}, n=−3

Exercise 122

(x−3)(x+2)=6

Exercise 123

The product of two consecutive integers is 156.

Answer

12 and 13; −13 and −12

Exercise 124

The area of a rectangular place mat is 168 square inches. Its length is two inches longer than the width. Find the length and width of the placemat.


This page titled Chapter 7 Review Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?