Contents

Preface
1Foundations
- Introduction
- 1.1Use the Language of Algebra
- 1.2Integers
- 1.3Fractions
- 1.4Decimals
- 1.5Properties of Real Numbers
- Chapter Review
  Key Terms
  Key Concepts
- Exercises
  Review Exercises
  Practice Test
2Solving Linear Equations
3Graphs and Functions
4Systems of Linear Equations
5Polynomials and Polynomial Functions
6Factoring
7Rational Expressions and Functions
8Roots and Radicals
9Quadratic Equations and Functions
10Exponential and Logarithmic Functions
11Conics
12Sequences, Series and Binomial Theorem
- Introduction
- 12.1Sequences
- 12.2Arithmetic Sequences
- 12.3Geometric Sequences and Series
- 12.4Binomial Theorem
- Chapter Review
  Key Terms
Answer Key
Index

Chapter 6

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6.1

$5 m^{2} 5 m^{2}$

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6.2

$7 x 7 x$

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6.3

$3 y^{2} (3 x + 2 x^{2} + 7 y) 3 y^{2} (3 x + 2 x^{2} + 7 y)$

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6.4

$3 p (p^{2} - 2 p q + 3 q^{2}) 3 p (p^{2} - 2 p q + 3 q^{2})$

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6.5

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

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6.6

$3 y^{2} (2 y - 5) 3 y^{2} (2 y - 5)$

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6.7

$3 x y (5 x^{2} - x y + 2 y^{2}) 3 x y (5 x^{2} - x y + 2 y^{2})$

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6.8

$2 a b (4 a^{2} + a b - 3 b^{2}) 2 a b (4 a^{2} + a b - 3 b^{2})$

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6.9

$−4 b (b^{2} - 4 b + 2) −4 b (b^{2} - 4 b + 2)$

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6.10

$−7 a (a^{2} - 3 a + 2) −7 a (a^{2} - 3 a + 2)$

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6.11

$(m + 3) (4 m - 7) (m + 3) (4 m - 7)$

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6.12

$(n - 4) (8 n + 5) (n - 4) (8 n + 5)$

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6.13

$(x + 8) (y + 3) (x + 8) (y + 3)$

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6.14

$(a + 7) (b + 8) (a + 7) (b + 8)$

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6.15

ⓐ $(x - 5) (x + 2) (x - 5) (x + 2)$
ⓑ $(5 x - 4) (4 x - 3) (5 x - 4) (4 x - 3)$

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6.16

ⓐ $(y + 4) (y - 7) (y + 4) (y - 7)$
ⓑ $(7 m - 3) (6 m - 5) (7 m - 3) (6 m - 5)$

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6.17

$(q + 4) (q + 6) (q + 4) (q + 6)$

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6.18

$(t + 2) (t + 12) (t + 2) (t + 12)$

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6.19

$(u - 3) (u - 6) (u - 3) (u - 6)$

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6.20

$(y - 7) (y - 9) (y - 7) (y - 9)$

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6.21

$(m + 3) (m + 6) (m + 3) (m + 6)$

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6.22

$(n - 3) (n - 4) (n - 3) (n - 4)$

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6.23

$(a - b) (a - 10 b) (a - b) (a - 10 b)$

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6.24

$(m - n) (m - 12 n) (m - n) (m - 12 n)$

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6.25

prime

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6.26

prime

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6.27

$5 x (x - 1) (x + 4) 5 x (x - 1) (x + 4)$

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6.28

$6 y (y - 2) (y + 5) 6 y (y - 2) (y + 5)$

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6.29

$(a + 1) (2 a + 3) (a + 1) (2 a + 3)$

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6.30

$(b + 1) (4 b + 1) (b + 1) (4 b + 1)$

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6.31

$(2 x - 3) (4 x - 1) (2 x - 3) (4 x - 1)$

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6.32

$(2 y - 7) (5 y - 1) (2 y - 7) (5 y - 1)$

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6.33

$(3 x + 2 y) (6 x - 5 y) (3 x + 2 y) (6 x - 5 y)$

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6.34

$(3 x + y) (10 x - 21 y) (3 x + y) (10 x - 21 y)$

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6.35

$5 n (n - 4) (3 n - 5) 5 n (n - 4) (3 n - 5)$

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6.36

$8 q (q + 6) (7 q - 2) 8 q (q + 6) (7 q - 2)$

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6.37

$(x + 2) (6 x + 1) (x + 2) (6 x + 1)$

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6.38

$(2 y + 1) (2 y + 3) (2 y + 1) (2 y + 3)$

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6.39

$4 (2 x - 3) (2 x - 1) 4 (2 x - 3) (2 x - 1)$

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6.40

$3 (3 w - 2) (2 w - 3) 3 (3 w - 2) (2 w - 3)$

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6.41

$(h^{2} - 2) (h^{2} + 6) (h^{2} - 2) (h^{2} + 6)$

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6.42

$(y^{2} + 4) (y^{2} - 5) (y^{2} + 4) (y^{2} - 5)$

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6.43

$(x - 3) (x - 1) (x - 3) (x - 1)$

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6.44

$(y - 1) (y + 1) (y - 1) (y + 1)$

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6.45

${(2 x + 3)}^{2} {(2 x + 3)}^{2}$

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6.46

${(3 y + 4)}^{2} {(3 y + 4)}^{2}$

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6.47

${(8 y - 5)}^{2} {(8 y - 5)}^{2}$

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6.48

${(4 z - 9)}^{2} {(4 z - 9)}^{2}$

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6.49

${(7 x + 6 y)}^{2} {(7 x + 6 y)}^{2}$

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6.50

${(8 m + 7 n)}^{2} {(8 m + 7 n)}^{2}$

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6.51

$2 y {(2 x - 3)}^{2} 2 y {(2 x - 3)}^{2}$

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6.52

$3 q {(3 p + 5)}^{2} 3 q {(3 p + 5)}^{2}$

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6.53

$(11 m - 1) (11 m + 1) (11 m - 1) (11 m + 1)$

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6.54

$(9 y - 1) (9 y + 1) (9 y - 1) (9 y + 1)$

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6.55

$(16 m - 5 n) (16 m + 5 n) (16 m - 5 n) (16 m + 5 n)$

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6.56

$(11 p - 3 q) (11 p + 3 q) (11 p - 3 q) (11 p + 3 q)$

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6.57

$2 y^{2} (x - 2) (x + 2) (x^{2} + 4) 2 y^{2} (x - 2) (x + 2) (x^{2} + 4)$

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6.58

$7 c^{2} (a - b) (a + b) (a^{2} + b^{2}) 7 c^{2} (a - b) (a + b) (a^{2} + b^{2})$

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6.59

$(x - 5 - y) (x - 5 + y) (x - 5 - y) (x - 5 + y)$

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6.60

$(x + 3 - 2 y) (x + 3 + 2 y) (x + 3 - 2 y) (x + 3 + 2 y)$

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6.61

$(x + 3) (x^{2} - 3 x + 9) (x + 3) (x^{2} - 3 x + 9)$

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6.62

$(y + 2) (y^{2} - 2 y + 4) (y + 2) (y^{2} - 2 y + 4)$

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6.63

$(2 x - 3 y) (4 x^{2} + 6 x y + 9 y^{2}) (2 x - 3 y) (4 x^{2} + 6 x y + 9 y^{2})$

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6.64

$(10 m - 5 n) (100 m^{2} + 50 m n + 25 n^{2}) (10 m - 5 n) (100 m^{2} + 50 m n + 25 n^{2})$

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6.65

$4 (5 p + q) (25 p^{2} - 5 p q + q^{2}) 4 (5 p + q) (25 p^{2} - 5 p q + q^{2})$

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6.66

$2 (6 c + 7 d) (36 c^{2} - 42 c d + 49 d^{2}) 2 (6 c + 7 d) (36 c^{2} - 42 c d + 49 d^{2})$

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6.67

$(−2 y + 1) (13 y^{2} + 5 y + 1) (−2 y + 1) (13 y^{2} + 5 y + 1)$

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6.68

$(−4 n + 3) (31 n^{2} + 21 n + 9) (−4 n + 3) (31 n^{2} + 21 n + 9)$

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6.69

$8 y (y - 1) (y + 3) 8 y (y - 1) (y + 3)$

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6.70

$5 y (y - 9) (y + 6) 5 y (y - 9) (y + 6)$

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6.71

$4 x (2 x - 3) (2 x + 3) 4 x (2 x - 3) (2 x + 3)$

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6.72

$3 (3 y - 4) (3 y + 4) 3 (3 y - 4) (3 y + 4)$

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6.73

${(2 x + 5 y)}^{2} {(2 x + 5 y)}^{2}$

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6.74

${(3 x - 4 y)}^{2} {(3 x - 4 y)}^{2}$

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6.75

$2 x y (25 x^{2} + 36) 2 x y (25 x^{2} + 36)$

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6.76

$3 x y (9 y^{2} + 16) 3 x y (9 y^{2} + 16)$

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6.77

$2 (5 m + 6 n) (25 m^{2} - 30 m n + 36 n^{2}) 2 (5 m + 6 n) (25 m^{2} - 30 m n + 36 n^{2})$

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6.78

$2 (p + 3 q) (p^{2} - 3 p q + 9 q^{2}) 2 (p + 3 q) (p^{2} - 3 p q + 9 q^{2})$

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6.79

$4 a b (a^{2} + 4) (a - 2) (a + 2) 4 a b (a^{2} + 4) (a - 2) (a + 2)$

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6.80

$7 x y (y^{2} + 1) (y - 1) (y + 1) 7 x y (y^{2} + 1) (y - 1) (y + 1)$

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6.81

$6 (x + b) (x - 2 c) 6 (x + b) (x - 2 c)$

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6.82

$2 (4 x - 1) (2 x + 3 y) 2 (4 x - 1) (2 x + 3 y)$

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6.83

$4 q (p - 3) (p - 1) 4 q (p - 3) (p - 1)$

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6.84

$3 p (2 q + 1) (q - 2) 3 p (2 q + 1) (q - 2)$

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6.85

$(2 x - 3 y - 5) (2 x - 3 y + 5) (2 x - 3 y - 5) (2 x - 3 y + 5)$

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6.86

$(4 x - 3 y - 8) (4 x - 3 y + 8) (4 x - 3 y - 8) (4 x - 3 y + 8)$

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6.87

$m = \frac{2}{3}, m = - \frac{1}{2} m = \frac{2}{3}, m = - \frac{1}{2}$

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6.88

$p = - \frac{3}{4}, p = \frac{3}{4} p = - \frac{3}{4}, p = \frac{3}{4}$

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6.89

$c = 2, c = \frac{4}{3} c = 2, c = \frac{4}{3}$

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6.90

$d = 3, d = - \frac{1}{2} d = 3, d = - \frac{1}{2}$

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6.91

$p = \frac{7}{5}, p = - \frac{7}{5} p = \frac{7}{5}, p = - \frac{7}{5}$

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6.92

$x = \frac{11}{6}, x = - \frac{11}{6} x = \frac{11}{6}, x = - \frac{11}{6}$

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6.93

$m = 1, m = \frac{3}{2} m = 1, m = \frac{3}{2}$

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6.94

$k = 3, k = −3 k = 3, k = −3$

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6.95

$a = - \frac{5}{2}, a = \frac{2}{3} a = - \frac{5}{2}, a = \frac{2}{3}$

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6.96

$b = −2, b = - \frac{1}{20} b = −2, b = - \frac{1}{20}$

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6.97

$x = 0, x = \frac{3}{2} x = 0, x = \frac{3}{2}$

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6.98

$y = 0, y = \frac{1}{4} y = 0, y = \frac{1}{4}$

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6.99

ⓐ $x = −3 x = −3$ or $x = 5 x = 5$
ⓑ $(−3, 7) (−3, 7)$ $(5, 7) (5, 7)$

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6.100

ⓐ $x = 1 x = 1$ or $x = 7 x = 7$
ⓑ $(1, −4) (1, −4)$ $(7, −4) (7, −4)$

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6.101

ⓐ $x = 1 x = 1$ or $x = \frac{5}{2} x = \frac{5}{2}$
ⓑ $(1, 0), (1, 0),$ $(\frac{5}{2}, 0) (\frac{5}{2}, 0)$ ⓒ $(0, 5) (0, 5)$

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6.102

ⓐ $x = −3 x = −3$ or $x = \frac{5}{6} x = \frac{5}{6}$
ⓑ $(−3, 0), (−3, 0),$ $(\frac{5}{6}, 0) (\frac{5}{6}, 0)$ ⓒ $(0, −15) (0, −15)$

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6.103

$−15, −17 −15, −17$ and 15, 17

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6.104

$−23, −21 −23, −21$ and 21, 23

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6.105

The width is 5 feet and length is 6 feet.

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6.106

The length of the patio is 12 feet and the width 15 feet.

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6.107

5 feet and 12 feet

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6.108

24 feet and 25 feet

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6.109

ⓐ 5 ⓑ 0;3 ⓒ 196

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6.110

ⓐ 4 ⓑ 0;2 ⓒ 144

Section 6.1 Exercises

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$2 p q 2 p q$

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$6 m^{2} n^{3} 6 m^{2} n^{3}$

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$2 a 2 a$

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$5 x^{3} y 5 x^{3} y$

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$3 (2 m + 3) 3 (2 m + 3)$

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11.

$9 (n - 7) 9 (n - 7)$

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13.

$3 (x^{2} + 2 x - 3) 3 (x^{2} + 2 x - 3)$

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15.

$2 (4 p^{2} + 2 p + 1) 2 (4 p^{2} + 2 p + 1)$

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17.

$8 y^{2} (y + 2) 8 y^{2} (y + 2)$

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19.

$5 x (x^{2} - 3 x + 4) 5 x (x^{2} - 3 x + 4)$

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21.

$3 x (8 x^{2} - 4 x + 5) 3 x (8 x^{2} - 4 x + 5)$

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23.

$6 y^{2} (2 x + 3 x^{2} - 5 y) 6 y^{2} (2 x + 3 x^{2} - 5 y)$

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25.

$4 x y (5 x^{2} - x y + 3 y^{2}) 4 x y (5 x^{2} - x y + 3 y^{2})$

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27.

$−2 (x + 2) −2 (x + 2)$

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29.

$−2 x (x^{2} - 9 x + 4) −2 x (x^{2} - 9 x + 4)$

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31.

$−4 p q (p^{2} + 3 p q - 4 q) −4 p q (p^{2} + 3 p q - 4 q)$

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33.

$(x + 1) (5 x + 3) (x + 1) (5 x + 3)$

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35.

$(b - 2) (3 b - 13) (b - 2) (3 b - 13)$

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37.

$(b + 5) (a + 3) (b + 5) (a + 3)$

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39.

$(y + 5) (8 y + 1) (y + 5) (8 y + 1)$

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41.

$(u + 2) (v - 9) (u + 2) (v - 9)$

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43.

$(u - 1) (u + 6) (u - 1) (u + 6)$

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45.

$(3 p - 5) (3 p + 4) (3 p - 5) (3 p + 4)$

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47.

$(n - 6) (m - 4) (n - 6) (m - 4)$

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49.

$(x - 7) (2 x - 5) (x - 7) (2 x - 5)$

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51.

$−9 x y (3 x + 2 y) −9 x y (3 x + 2 y)$

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53.

$(x^{2} + 2) (3 x - 7) (x^{2} + 2) (3 x - 7)$

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55.

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

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57.

Answers will vary.

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59.

Answers will vary.

Section 6.2 Exercises

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61.

$(p + 5) (p + 6) (p + 5) (p + 6)$

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63.

$(n + 3) (n + 16) (n + 3) (n + 16)$

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65.

$(a + 5) (a + 20) (a + 5) (a + 20)$

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67.

$(x - 2) (x - 6) (x - 2) (x - 6)$

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69.

$(y - 3) (y - 15) (y - 3) (y - 15)$

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71.

$(x - 1) (x - 7) (x - 1) (x - 7)$

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73.

$(p - 1) (p + 6) (p - 1) (p + 6)$

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75.

$(x - 4) (x - 2) (x - 4) (x - 2)$

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77.

$(x - 12) (x + 1) (x - 12) (x + 1)$

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79.

$(x + 8 y) (x - 10 y) (x + 8 y) (x - 10 y)$

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81.

$(m + n) (m - 65 n) (m + n) (m - 65 n)$

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83.

$(a + 8 b) (a - 3 b) (a + 8 b) (a - 3 b)$

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85.

Prime

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87.

Prime

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89.

$p (p - 10) (p + 2) p (p - 10) (p + 2)$

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91.

$3 m (m - 5) (m - 2) 3 m (m - 5) (m - 2)$

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93.

$5 x^{2} (x - 3) (x + 5) 5 x^{2} (x - 3) (x + 5)$

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95.

$(2 t + 5) (t + 1) (2 t + 5) (t + 1)$

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97.

$(11 x + 1) (x + 3) (11 x + 1) (x + 3)$

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99.

$(4 w - 1) (w - 1) (4 w - 1) (w - 1)$

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101.

$(4 q + 1) (q - 2) (4 q + 1) (q - 2)$

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103.

$(2 p - 5 q) (3 p - 2 q) (2 p - 5 q) (3 p - 2 q)$

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105.

$(4 a - 3 b) (a + 5 b) (4 a - 3 b) (a + 5 b)$

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107.

$−16 (x + 1) (x + 1) −16 (x + 1) (x + 1)$

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109.

$−10 q (3 q + 2) (q + 4) −10 q (3 q + 2) (q + 4)$

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111.

$(5 n + 1) (n + 4) (5 n + 1) (n + 4)$

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113.

$(2 k - 3) (2 k - 5) (2 k - 3) (2 k - 5)$

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115.

$(3 y + 5) (2 y - 3) (3 y + 5) (2 y - 3)$

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117.

$(2 n + 3) (n - 15) (2 n + 3) (n - 15)$

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119.

$10 (6 y - 1) (y + 5) 10 (6 y - 1) (y + 5)$

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121.

$3 z (8 z + 3) (2 z - 5) 3 z (8 z + 3) (2 z - 5)$

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123.

$8 (2 s + 3) (s + 1) 8 (2 s + 3) (s + 1)$

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125.

$12 (4 y - 3) (y + 1) 12 (4 y - 3) (y + 1)$

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127.

$(x^{2} + 3) (x^{2} - 4) (x^{2} + 3) (x^{2} - 4)$

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129.

$(x^{2} - 7) (x^{2} + 4) (x^{2} - 7) (x^{2} + 4)$

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131.

$(x - 12) (x + 1) (x - 12) (x + 1)$

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133.

$(3 y - 4) (3 y - 1) (3 y - 4) (3 y - 1)$

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135.

$(u - 6) (u - 6) (u - 6) (u - 6)$

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137.

$(r - 4 s) (r - 16 s) (r - 4 s) (r - 16 s)$

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139.

$(4 y - 7) (3 y - 2) (4 y - 7) (3 y - 2)$

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141.

$(2 n - 1) (3 n + 4) (2 n - 1) (3 n + 4)$

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143.

$13 (z^{2} + 3 z - 2) 13 (z^{2} + 3 z - 2)$

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145.

$3 p (p + 7) 3 p (p + 7)$

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147.

$6 (r + 2) (r + 3) 6 (r + 2) (r + 3)$

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149.

$4 (2 n + 1) (3 n + 1) 4 (2 n + 1) (3 n + 1)$

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151.

$(x^{2} + 2) (x^{2} - 6) (x^{2} + 2) (x^{2} - 6)$

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153.

$(x - 9) (x + 6) (x - 9) (x + 6)$

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155.

Answers will vary.

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157.

Answers will vary.

Section 6.3 Exercises

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159.

${(4 y + 3)}^{2} {(4 y + 3)}^{2}$

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161.

${(6 s + 7)}^{2} {(6 s + 7)}^{2}$

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163.

${(10 x - 1)}^{2} {(10 x - 1)}^{2}$

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165.

${(5 n - 12)}^{2} {(5 n - 12)}^{2}$

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167.

${(7 x + 2 y)}^{2} {(7 x + 2 y)}^{2}$

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169.

$(10 y - 1)^{2} (10 y - 1)^{2}$

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171.

$10 j {(k + 4)}^{2} 10 j {(k + 4)}^{2}$

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173.

$3 u^{2} {(5 u - v)}^{2} 3 u^{2} {(5 u - v)}^{2}$

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175.

$(5 v - 1) (5 v + 1) (5 v - 1) (5 v + 1)$

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177.

$(2 - 7 x) (2 + 7 x) (2 - 7 x) (2 + 7 x)$

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179.

$6 p^{2} (q - 3) (q + 3) 6 p^{2} (q - 3) (q + 3)$

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181.

$6 (4 p^{2} + 9) 6 (4 p^{2} + 9)$

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183.

$(11 x - 12 y) (11 x + 12 y) (11 x - 12 y) (11 x + 12 y)$

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185.

$(13 c - 6 d) (13 c + 6 d) (13 c - 6 d) (13 c + 6 d)$

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187.

$(2 z - 1) (2 z + 1) (4 z^{2} + 1) (2 z - 1) (2 z + 1) (4 z^{2} + 1)$

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189.

$2 b^{2} (3 a - 2) (3 a + 2) (9 a^{2} + 4) 2 b^{2} (3 a - 2) (3 a + 2) (9 a^{2} + 4)$

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191.

$(x - 8 - y) (x - 8 + y) (x - 8 - y) (x - 8 + y)$

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193.

$(a + 3 - 3 b) (a + 3 + 3 b) (a + 3 - 3 b) (a + 3 + 3 b)$

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195.

$(x + 5) (x^{2} - 5 x + 25) (x + 5) (x^{2} - 5 x + 25)$

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197.

$(z^{2} - 3) (z^{4} + 3 z^{2} + 9) (z^{2} - 3) (z^{4} + 3 z^{2} + 9)$

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199.

$(2 - 7 t) (4 + 14 t + 49 t^{2}) (2 - 7 t) (4 + 14 t + 49 t^{2})$

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201.

$(2 y - 5 z) (4 y^{2} + 10 y z + 25 z^{2}) (2 y - 5 z) (4 y^{2} + 10 y z + 25 z^{2})$

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203.

$(6 a + 5 b) (36 a^{2} - 30 a b + 25 b^{2}) (6 a + 5 b) (36 a^{2} - 30 a b + 25 b^{2})$

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205.

$7 (k + 2) (k^{2} - 2 k + 4) 7 (k + 2) (k^{2} - 2 k + 4)$

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207.

$2 x^{2} (1 - 2 y) (1 + 2 y + 4 y^{2}) 2 x^{2} (1 - 2 y) (1 + 2 y + 4 y^{2})$

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209.

$9 (x + 1) (x^{2} + 3) 9 (x + 1) (x^{2} + 3)$

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211.

$− (3 y + 5) (21 y^{2} - 30 y + 25) − (3 y + 5) (21 y^{2} - 30 y + 25)$

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213.

$(8 a - 5) (8 a + 5) (8 a - 5) (8 a + 5)$

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215.

$3 (3 q - 1) (3 q + 1) 3 (3 q - 1) (3 q + 1)$

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217.

${(4 x - 9)}^{2} {(4 x - 9)}^{2}$

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219.

$2 (4 p^{2} + 1) 2 (4 p^{2} + 1)$

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221.

$(5 - 2 y) (25 + 10 y + 4 y^{2}) (5 - 2 y) (25 + 10 y + 4 y^{2})$

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223.

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

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225.

$(x + y - 5) (x - y - 5) (x + y - 5) (x - y - 5)$

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227.

$(3 x + 1) (3 x^{2} + 1) (3 x + 1) (3 x^{2} + 1)$

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229.

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231.

Answers will vary.

Section 6.4 Exercises

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233.

$(2 n - 1) (n + 7) (2 n - 1) (n + 7)$

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235.

$a^{3} (a^{2} + 9) a^{3} (a^{2} + 9)$

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237.

$(11 r - s) (11 r + s) (11 r - s) (11 r + s)$

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239.

$8 (m - 2) (m + 2) 8 (m - 2) (m + 2)$

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241.

${(5 w - 6)}^{2} {(5 w - 6)}^{2}$

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243.

${(m + 7 n)}^{2} {(m + 7 n)}^{2}$

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245.

$7 (b + 3) (b - 2) 7 (b + 3) (b - 2)$

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247.

$3 x y (x - 3) (x^{2} + 3 x + 9) 3 x y (x - 3) (x^{2} + 3 x + 9)$

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249.

$(k - 2) (k + 2) (k^{2} + 4) (k - 2) (k + 2) (k^{2} + 4)$

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251.

$5 x y^{2} (x^{2} + 4) (x + 2) (x - 2) 5 x y^{2} (x^{2} + 4) (x + 2) (x - 2)$

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253.

$3 (5 p + 4) (q - 1) 3 (5 p + 4) (q - 1)$

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255.

$4 (x + 3) (x + 7) 4 (x + 3) (x + 7)$

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257.

$4 u^{2} (u + 1) (u^{2} - u + 1) 4 u^{2} (u + 1) (u^{2} - u + 1)$

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259.

prime

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261.

$10 (m - 5) (m + 5) (m^{2} + 25) 10 (m - 5) (m + 5) (m^{2} + 25)$

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263.

$3 y (3 x + 2) (4 x - 1) 3 y (3 x + 2) (4 x - 1)$

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265.

$(2 x - 3 y) (4 x^{2} + 6 x y + 9 y^{2}) (2 x - 3 y) (4 x^{2} + 6 x y + 9 y^{2})$

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267.

$(y + 1) (y - 1) (y^{2} - y + 1) (y^{2} + y + 1) (y + 1) (y - 1) (y^{2} - y + 1) (y^{2} + y + 1)$

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269.

$(3 x - y + 7) (3 x - y - 7) (3 x - y + 7) (3 x - y - 7)$

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271.

$(3 x - 2)^{2} (3 x - 2)^{2}$

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273.

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275.

Answers will vary.

Section 6.5 Exercises

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277.

$a = 10 / 3, a = 7 / 2 a = 10 / 3, a = 7 / 2$

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279.

$m = 0, m = 5 / 12 m = 0, m = 5 / 12$

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281.

$x = 1 / 2 x = 1 / 2$

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283.

$a = \frac{−4}{5}, a = 6 a = \frac{−4}{5}, a = 6$

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285.

$m = 5 / 4, m = 3 m = 5 / 4, m = 3$

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287.

$a = −1, a = 0 a = −1, a = 0$

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289.

$m = 12 / 7, m = −12 / 7 m = 12 / 7, m = −12 / 7$

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291.

$y = −9 / 4, y = 9 / 4 y = −9 / 4, y = 9 / 4$

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293.

$n = −6 / 11, n = 6 / 11 n = −6 / 11, n = 6 / 11$

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295.

$x = 2, x = −5 x = 2, x = −5$

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297.

$x = 3 / 2, x = −1 x = 3 / 2, x = −1$

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299.

$x = 2, x = −4 / 3 x = 2, x = −4 / 3$

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301.

$x = 3 / 2 x = 3 / 2$

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303.

$x = 2, x = −4 / 3 x = 2, x = −4 / 3$

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305.

$x = −3 / 2, x = 1 / 3 x = −3 / 2, x = 1 / 3$

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307.

$p = 0, p = ¾ p = 0, p = ¾$

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309.

$x = 0, x = 6 x = 0, x = 6$

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311.

$x = 0, x = –1 / 3 x = 0, x = –1 / 3$

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313.

ⓐ $x = 2 x = 2$ or $x = 6 x = 6$ ⓑ $(2, −4) (2, −4)$ $(6, −4) (6, −4)$

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315.

ⓐ $x = \frac{3}{2} x = \frac{3}{2}$ or $x = \frac{3}{4} x = \frac{3}{4}$
ⓑ $(\frac{3}{2}, −4) (\frac{3}{2}, −4)$ $(\frac{3}{4}, −4) (\frac{3}{4}, −4)$

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317.

ⓐ $x = \frac{2}{3} x = \frac{2}{3}$ or $x = - \frac{2}{3} x = - \frac{2}{3}$
ⓑ $(\frac{2}{3}, 0) (\frac{2}{3}, 0)$ , $(- \frac{2}{3}, 0) (- \frac{2}{3}, 0)$ ⓒ $(0, −4) (0, −4)$

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319.

ⓐ $x = \frac{5}{3} x = \frac{5}{3}$ or $x = - \frac{1}{2} x = - \frac{1}{2}$
ⓑ $(\frac{5}{3}, 0) (\frac{5}{3}, 0)$ , $(- \frac{1}{2}, 0) (- \frac{1}{2}, 0)$ ⓒ $(0, −5) (0, −5)$

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321.

$−13, −11 −13, −11$ and 11, 13

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323.

$−14, −12 −14, −12$ and 12, 14

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325.

$44$ and 7

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327.

5, 11

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329.

6, 8

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331.

8, 15, 17

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333.

ⓐ 0, 2 ⓑ 1

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335.

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Review Exercises

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337.

$3 a b^{2} 3 a b^{2}$

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339.

$3 y 3 y$

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341.

$7 (5 y + 12) 7 (5 y + 12)$

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343.

$3 x (6 x^{2} - 5) 3 x (6 x^{2} - 5)$

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345.

$4 x (x^{2} - 3 x + 4) 4 x (x^{2} - 3 x + 4)$

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347.

$−3 x (x^{2} - 9 x + 4) −3 x (x^{2} - 9 x + 4)$

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349.

$(a + b) (x - y) (a + b) (x - y)$

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351.

$(x - 3) (x + 7) (x - 3) (x + 7)$

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353.

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

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355.

$(a + 3) (a + 11) (a + 3) (a + 11)$

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357.

$(m + 9) (m - 6) (m + 9) (m - 6)$

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359.

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

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361.

$(a + 7 b) (a - 3 b) (a + 7 b) (a - 3 b)$

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363.

Prime

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365.

$3 y (y - 5) (y - 2) 3 y (y - 5) (y - 2)$

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367.

$(5 y + 9) (y + 1) (5 y + 9) (y + 1)$

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369.

$(5 y + 1) (2 y - 11) (5 y + 1) (2 y - 11)$

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371.

$−9 (9 a - 1) (a + 2) −9 (9 a - 1) (a + 2)$

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373.

$(3 a - 1) (6 a - 1) (3 a - 1) (6 a - 1)$

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375.

$(3 x - 1) (5 x + 2) (3 x - 1) (5 x + 2)$

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377.

$3 (x + 4) (x - 3) 3 (x + 4) (x - 3)$

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379.

$3 (2 a - 7) (3 a + 1) 3 (2 a - 7) (3 a + 1)$

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381.

$(x^{2} - 15) (x^{2} + 2) (x^{2} - 15) (x^{2} + 2)$

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383.

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

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385.

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

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387.

$3 u^{2} {(5 u - v)}^{2} 3 u^{2} {(5 u - v)}^{2}$

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389.

$(13 m + n) (13 m - n) (13 m + n) (13 m - n)$

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391.

$(3 + 11 y) (3 - 11 y) (3 + 11 y) (3 - 11 y)$

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393.

$n (13 n + 1) (13 n - 1) n (13 n + 1) (13 n - 1)$

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395.

$6 (4 p^{2} + 9) 6 (4 p^{2} + 9)$

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397.

$(2 z - 1) (2 z + 1) (4 z^{2} + 1) (2 z - 1) (2 z + 1) (4 z^{2} + 1)$

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399.

$(a + 3 - 3 b) (a + 3 + 3 b) (a + 3 - 3 b) (a + 3 + 3 b)$

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401.

$(a - 5) (a^{2} + 5 a + 25) (a - 5) (a^{2} + 5 a + 25)$

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403.

$2 (m + 3) (m^{2} - 3 m + 9) 2 (m + 3) (m^{2} - 3 m + 9)$

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405.

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

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407.

${(4 n - 7 m)}^{2} {(4 n - 7 m)}^{2}$

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409.

$5 u^{2} (u + 3) (u - 3) 5 u^{2} (u + 3) (u - 3)$

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411.

prime

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413.

$(b - 4) (b^{2} + 4 b + 16) (b - 4) (b^{2} + 4 b + 16)$

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415.

$(2 b + 5 c) (b - c) (2 b + 5 c) (b - c)$

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417.

$5 (q + 3) (q - 6) 5 (q + 3) (q - 6)$

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419.

$10 (m - 5) (m + 5) (m^{2} + 25) 10 (m - 5) (m + 5) (m^{2} + 25)$

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421.

$(4 x - 3 y + 8) (4 x - 3 y - 8) (4 x - 3 y + 8) (4 x - 3 y - 8)$

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423.

$b = −1 / 5, b = −1 / 6 b = −1 / 5, b = −1 / 6$

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425.

$x = 1 / 2 x = 1 / 2$

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427.

$x = −4, x = −5 x = −4, x = −5$

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429.

$p = - \frac{5}{2}, p = 8 p = - \frac{5}{2}, p = 8$

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431.

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

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433.

$x = 2, x = −5 x = 2, x = −5$

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435.

$p = 0, p = ¾ p = 0, p = ¾$

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437.

ⓐ $x = −7 x = −7$ or $x = −4 x = −4$
ⓑ $(−7, −8) (−7, −8)$ $(−4, −8) (−4, −8)$

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439.

ⓐ $x = \frac{7}{8} x = \frac{7}{8}$ or $x = - \frac{7}{8} x = - \frac{7}{8}$
ⓑ $(\frac{7}{8}, 0), (\frac{7}{8}, 0),$ $(- \frac{7}{8}, 0) (- \frac{7}{8}, 0)$ ⓒ $(0, −49) (0, −49)$