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Mathematics LibreTexts

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.

    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.