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 5

Try It

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5.1

ⓐ monomial, 0
ⓑ polynomial, 3 ⓒ trinomial, 3
ⓓ binomial, 2 ⓔ monomial, 10

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5.2

ⓐbinomial, 3 ⓑ trinomial, 3 ⓒ monomial, 0 ⓓ polynomial, 4 ⓔ monomial, 7

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5.3

ⓐ $21 q^{2} 21 q^{2}$ ⓑ $13 m n^{3} 13 m n^{3}$

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5.4

ⓐ $−7 c^{2} −7 c^{2}$ ⓑ $−10 y^{2} z^{3} −10 y^{2} z^{3}$

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5.5

ⓐ $5 y^{2} + 3 z^{2} 5 y^{2} + 3 z^{2}$
ⓑ $m^{2} n^{2} - 8 m^{2} + 4 n^{2} m^{2} n^{2} - 8 m^{2} + 4 n^{2}$

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5.6

ⓐ $−4 m^{2} + n^{2} −4 m^{2} + n^{2}$
ⓑ $p q^{2} - 6 p - 5 q^{2} p q^{2} - 6 p - 5 q^{2}$

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5.7

$8 x^{2} - 11 x + 8 8 x^{2} - 11 x + 8$

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5.8

$17 y^{2} + 14 y + 1 17 y^{2} + 14 y + 1$

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5.9

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

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5.10

$6 b^{2} + 3 6 b^{2} + 3$

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5.11

$−5 a b + 7 b^{2} −5 a b + 7 b^{2}$

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5.12

$7 m n + 4 n^{2} 7 m n + 4 n^{2}$

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5.13

$5 x^{2} - 5 x y + 5 y^{2} 5 x^{2} - 5 x y + 5 y^{2}$

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5.14

$7 x^{2} - 6 x y - 2 y^{2} 7 x^{2} - 6 x y - 2 y^{2}$

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5.15

$x^{3} + y^{3} x^{3} + y^{3}$

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5.16

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

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5.17

ⓐ 18 ⓑ 50 ⓒ $−15 −15$

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5.18

ⓐ 20 ⓑ 2 ⓒ $−4 −4$

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5.19

The height is $150150$ feet.

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5.20

The height is 31 feet.

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5.21

ⓐ $(f + g) (x) = 3 x^{2} - 6 x - 3 (f + g) (x) = 3 x^{2} - 6 x - 3$ ⓑ $(f + g) (3) = 6 (f + g) (3) = 6$
ⓒ $(f - g) (x) = x^{2} - 2 x + 9 (f - g) (x) = x^{2} - 2 x + 9$
ⓓ $(f - g) (−2) = 17 (f - g) (−2) = 17$

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5.22

ⓐ $(f + g) (x) = 6 x^{2} - x + 7 (f + g) (x) = 6 x^{2} - x + 7$ ⓑ $(f + g) (3) = 58 (f + g) (3) = 58$
ⓒ $(f - g) (x) = 4 x^{2} - 7 x - 9 (f - g) (x) = 4 x^{2} - 7 x - 9$
ⓓ $(f - g) (−2) = 21 (f - g) (−2) = 21$

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5.23

ⓐ $b^{17} b^{17}$ ⓑ $4^{3 x} 4^{3 x}$ ⓒ $12 p^{6} 12 p^{6}$
ⓓ $x^{18} x^{18}$

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5.24

ⓐ $x^{16} x^{16}$ ⓑ $10^{x + 1} 10^{x + 1}$ ⓐ $12 z^{8} 12 z^{8}$
ⓓ $b^{19} b^{19}$

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5.25

ⓐ $x^{5} x^{5}$ ⓑ $6^{9} 6^{9}$ ⓒ $\frac{1}{x^{4}} \frac{1}{x^{4}}$
ⓓ $\frac{1}{12^{15}} \frac{1}{12^{15}}$

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5.26

ⓐ $y^{6} y^{6}$ ⓑ $10^{8} 10^{8}$ ⓒ $\frac{1}{m^{8}} \frac{1}{m^{8}}$
ⓓ $\frac{1}{9^{11}} \frac{1}{9^{11}}$

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5.27

ⓐ 1 ⓑ 1

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5.28

ⓐ 1 ⓑ 1

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5.29

ⓐ $\frac{1}{z^{3}} \frac{1}{z^{3}}$ ⓑ $\frac{1}{10^{7}} \frac{1}{10^{7}}$ ⓒ $p^{8} p^{8}$ ⓓ $6464$

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5.30

ⓐ $\frac{1}{n^{2}} \frac{1}{n^{2}}$ ⓑ $\frac{1}{10,000} \frac{1}{10,000}$ ⓒ $q^{7} q^{7}$
ⓓ $1616$

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5.31

ⓐ $\frac{81}{16} \frac{81}{16}$ ⓑ $\frac{n^{2}}{m^{2}} \frac{n^{2}}{m^{2}}$

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5.32

ⓐ $\frac{125}{27} \frac{125}{27}$ ⓑ $\frac{b^{4}}{a^{4}} \frac{b^{4}}{a^{4}}$

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5.33

ⓐ $\frac{1}{z^{9}} \frac{1}{z^{9}}$ ⓑ $\frac{1}{p^{3} q^{3}} \frac{1}{p^{3} q^{3}}$ ⓒ $- \frac{12 v^{5}}{u} - \frac{12 v^{5}}{u}$

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5.34

ⓐ $\frac{1}{c^{15}} \frac{1}{c^{15}}$ ⓑ $\frac{1}{r^{2} s^{8}} \frac{1}{r^{2} s^{8}}$ ⓒ $\frac{30 d^{3}}{c^{8}} \frac{30 d^{3}}{c^{8}}$

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5.35

ⓐ $b^{35} b^{35}$ ⓑ $5^{12} 5^{12}$ ⓒ $a^{48} a^{48}$

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5.36

ⓐ $z^{54} z^{54}$ ⓑ $3^{49} 3^{49}$ ⓒ $q^{29} q^{29}$

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5.37

ⓐ $32 w^{5} x^{5} 32 w^{5} x^{5}$ ⓑ 1 ⓒ $\frac{1}{16 b^{12}} \frac{1}{16 b^{12}}$
ⓓ $\frac{64}{a^{8}} \frac{64}{a^{8}}$

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5.38

ⓐ $−27 y^{3} −27 y^{3}$ ⓑ 1 ⓒ $\frac{1}{16 x^{8}} \frac{1}{16 x^{8}}$
ⓓ $\frac{8}{c^{12}} \frac{8}{c^{12}}$

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5.39

ⓐ $\frac{p^{4}}{10000} \frac{p^{4}}{10000}$ ⓑ $\frac{n^{7}}{m^{7}} \frac{n^{7}}{m^{7}}$
ⓒ $\frac{81 a^{4} b^{12}}{c^{8}} \frac{81 a^{4} b^{12}}{c^{8}}$ ⓓ $\frac{27}{x^{6} y^{9}} \frac{27}{x^{6} y^{9}}$

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5.40

ⓐ $\frac{−8}{q^{3}} \frac{−8}{q^{3}}$ ⓑ $\frac{x^{4}}{w^{4}} \frac{x^{4}}{w^{4}}$ ⓒ $\frac{x^{2} y^{6}}{9 z^{4}} \frac{x^{2} y^{6}}{9 z^{4}}$
ⓓ $\frac{8 n^{6}}{m^{6}} \frac{8 n^{6}}{m^{6}}$

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5.41

ⓐ $81 c^{24} d^{30} 81 c^{24} d^{30}$ ⓑ $\frac{1}{a^{18}} \frac{1}{a^{18}}$
ⓒ $y^{15} y^{15}$

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5.42

ⓐ $256 a^{22} b^{24} 256 a^{22} b^{24}$ ⓑ $\frac{1}{p^{39}} \frac{1}{p^{39}}$
ⓒ $2 x^{3} y^{10} 2 x^{3} y^{10}$

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5.43

ⓐ $9.6 \times 10^{4} 9.6$ ⓑ $7.8 \times 10^{−3} 7.8$

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5.44

ⓐ $4.83 \times 10^{4} 4.83$
ⓑ $1.29 \times 10^{−2} 1.29$

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5.45

ⓐ 1,300 ⓑ $−0.00012 −0.00012$

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5.46

ⓐ $−950,000 −950,000$ ⓑ 0.075

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5.47

ⓐ $−0.006 −0.006$ ⓑ 20,000

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5.48

ⓐ $−0.009 −0.009$ ⓑ 400,000

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5.49

ⓐ $−35 y^{11} −35 y^{11}$ ⓑ $6 a^{5} b^{6} 6 a^{5} b^{6}$

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5.50

ⓐ $54 b^{9} 54 b^{9}$ ⓑ $8 r^{11} s^{8} 8 r^{11} s^{8}$

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5.51

ⓐ $−15 y^{3} - 24 y^{2} + 21 y −15 y^{3} - 24 y^{2} + 21 y$
ⓑ $12 x^{4} y^{2} - 20 x^{3} y^{3} + 12 x^{2} y^{4} 12 x^{4} y^{2} - 20 x^{3} y^{3} + 12 x^{2} y^{4}$

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5.52

ⓐ $8 x^{4} - 24 x^{3} + 20 x^{2} 8 x^{4} - 24 x^{3} + 20 x^{2}$
ⓑ $−18 a^{5} b + 12 a^{4} b^{2} - 36 a^{3} b^{3} −18 a^{5} b + 12 a^{4} b^{2} - 36 a^{3} b^{3}$

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5.53

ⓐ $x^{2} + 17 x + 72 x^{2} + 17 x + 72$
ⓑ $15 c^{2} + 14 c - 8 15 c^{2} + 14 c - 8$

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5.54

ⓐ $20 x^{2} + 51 x + 27 20 x^{2} + 51 x + 27$
ⓑ $30 y^{2} - 3 y - 6 30 y^{2} - 3 y - 6$

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5.55

ⓐ $x^{2} - 2 x - 35 x^{2} - 2 x - 35$
ⓑ $15 x^{2} + 29 x - 14 15 x^{2} + 29 x - 14$

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5.56

ⓐ $b^{2} + 3 b - 18 b^{2} + 3 b - 18$
ⓑ $16 y^{2} - 20 y - 50 16 y^{2} - 20 y - 50$

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5.57

ⓐ $x^{3} - 8 x^{2} + 6 x - 48 x^{3} - 8 x^{2} + 6 x - 48$
ⓑ $8 a^{2} b^{2} + 12 a b - 20 8 a^{2} b^{2} + 12 a b - 20$

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5.58

ⓐ $y^{3} - 9 y^{2} + 7 y - 63 y^{3} - 9 y^{2} + 7 y - 63$
ⓑ $8 x^{2} y^{2} + 2 x y - 15 8 x^{2} y^{2} + 2 x y - 15$

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5.59

$15 m^{2} - 51 m + 42 15 m^{2} - 51 m + 42$

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5.60

$42 b^{2} - 53 b + 15 42 b^{2} - 53 b + 15$

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5.61

ⓐ $y^{3} - 8 y^{2} + 17 y - 6 y^{3} - 8 y^{2} + 17 y - 6$
ⓑ $y^{3} - 8 y^{2} + 17 y - 6 y^{3} - 8 y^{2} + 17 y - 6$

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5.62

ⓐ $2 x^{3} + 5 x^{2} - 7 x + 20 2 x^{3} + 5 x^{2} - 7 x + 20$
ⓑ $y^{3} - 8 y^{2} + 17 y - 6 y^{3} - 8 y^{2} + 17 y - 6$

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5.63

ⓐ $x^{2} + 18 x + 81 x^{2} + 18 x + 81$
ⓑ $4 c^{2} - 4 c d + d^{2} 4 c^{2} - 4 c d + d^{2}$

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5.64

ⓐ $y^{2} + 22 y + 121 y^{2} + 22 y + 121$
ⓑ $16 x^{2} - 40 x y + 25 y^{2} 16 x^{2} - 40 x y + 25 y^{2}$

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5.65

ⓐ $36 x^{2} - 25 36 x^{2} - 25$
ⓑ $16 p^{2} - 49 q^{2} 16 p^{2} - 49 q^{2}$

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5.66

ⓐ $4 x^{2} - 49 4 x^{2} - 49$ ⓑ $9 x^{2} - y^{2} 9 x^{2} - y^{2}$

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5.67

ⓐ FOIL; $18 b^{2} + 77 b - 18 18 b^{2} + 77 b - 18$
ⓑ Binomial Squares; $81 p^{2} - 72 p + 16 81 p^{2} - 72 p + 16$
ⓒ Binomial Squares; $49 y^{2} + 14 y + 1 49 y^{2} + 14 y + 1$
ⓓ Product of Conjugates; $16 r^{2} - 9 16 r^{2} - 9$

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5.68

ⓐ Binomial Squares; $36 x^{2} + 84 x + 49 36 x^{2} + 84 x + 49$ ⓑ Product of Conjugates; $9 x^{2} - 16 9 x^{2} - 16$ ⓒ FOIL; $10 x^{2} - 29 x + 10 10 x^{2} - 29 x + 10$ ⓓ Binomial Squares; $36 n^{2} - 12 n + 1 36 n^{2} - 12 n + 1$

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5.69

ⓐ $(f \cdot g) (x) = x^{3} - 7 x^{2} + 13 x - 15 (f \cdot g) (x) = x^{3} - 7 x^{2} + 13 x - 15$
ⓑ $(f \cdot g) (2) = −9 (f \cdot g) (2) = −9$

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5.70

ⓐ $(f \cdot g) (x) = x^{3} + x^{2} - 52 x - 28 (f \cdot g) (x) = x^{3} + x^{2} - 52 x - 28$
ⓑ $(f \cdot g) (2) = −120 (f \cdot g) (2) = −120$

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5.71

$- \frac{9}{a^{5} b} - \frac{9}{a^{5} b}$

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5.72

$\frac{−9 d}{c^{4}} \frac{−9 d}{c^{4}}$

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5.73

$\frac{4 y^{2}}{7 x^{4}} \frac{4 y^{2}}{7 x^{4}}$

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5.74

$\frac{5}{8 m^{5} n^{3}} \frac{5}{8 m^{5} n^{3}}$

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5.75

$−4 a + 2 b −4 a + 2 b$

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5.76

$8 a^{5} b + 6 a^{3} b^{2} 8 a^{5} b + 6 a^{3} b^{2}$

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5.77

$y + 7 y + 7$

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5.78

$m + 5 m + 5$

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5.79

$x^{3} - 3 x^{2} + 2 x + 1 + \frac{3}{x + 3} x^{3} - 3 x^{2} + 2 x + 1 + \frac{3}{x + 3}$

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5.80

$x^{3} - 3 x^{2} - 2 x - 1 - \frac{3}{x + 3} x^{3} - 3 x^{2} - 2 x - 1 - \frac{3}{x + 3}$

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5.81

$x^{2} + 4 x + 16 x^{2} + 4 x + 16$

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5.82

$25 x^{2} + 10 x + 4 25 x^{2} + 10 x + 4$

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5.83

$3 x^{2} + 4 x - 2; 2 3 x^{2} + 4 x - 2;$

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5.84

$4 x^{2} - 3 x + 1; 1 4 x^{2} - 3 x + 1;$

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5.85

$x^{3} - 4 x^{2} + 5; 0 x^{3} - 4 x^{2} + 5;$

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5.86

$x^{3} - 3 x^{2} + 2; 0 x^{3} - 3 x^{2} + 2;$

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5.87

ⓐ $(\frac{f}{g}) (x) = x - 8 (\frac{f}{g}) (x) = x - 8$
ⓑ $(\frac{f}{g}) (−3) = −11 (\frac{f}{g}) (−3) = −11$

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5.88

ⓐ $(\frac{f}{g}) (x) = x - 9 (\frac{f}{g}) (x) = x - 9$
ⓑ $(\frac{f}{g}) (−5) = −14 (\frac{f}{g}) (−5) = −14$

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5.89

$−1 −1$

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5.90

$66$

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5.91

yes

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5.92

yes

Section 5.1 Exercises

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ⓐ trinomial, 5 ⓑ polynomial, 3 ⓒ binomial, 2 ⓓ monomial, 0
ⓔ binomial, 1

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ⓐ binomial ⓑ trinomial
ⓒ polynomial ⓓ trinomial
ⓔ monomial

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ⓐ $trinomial, 2 trinomial, 2$ ⓑ $monomial, 4 monomial, 4$ ⓒ $binomial, 1 binomial, 1$ ⓓ $quadrinomial, 3 quadrinomial, 3$
ⓔ $monomial, 0 monomial, 0$

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ⓐ $binomial, 1 binomial, 1$ ⓑ $trinomial, 2 trinomial, 2$ ⓒ $pentanomial, 3 pentanomial, 3$ ⓓ $binomial, 3 binomial, 3$
ⓔ $monomial, 0 monomial, 0$

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ⓐ ${12x}^{2} {12x}^{2}$ ⓑ $− 5a −$

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11.

ⓐ $6 w 6 w$ ⓑ $19 x^{2} y 19 x^{2} y$

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13.

${12x}^{2} - 5a {12x}^{2} -$

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15.

$6 w + 19 x^{2} y 6 w + 19 x^{2} y$

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17.

ⓐ $−22 b −22 b$ ⓑ $16 x y 16 x y$

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19.

ⓐ $−10 a + 5 b −10 a + 5 b$
ⓑ $p q^{2} - 4 p - 3 q^{2} p q^{2} - 4 p - 3 q^{2}$

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21.

ⓐ $−4 a^{2} + b^{2} −4 a^{2} + b^{2}$
ⓑ $x^{2} y - 3 x + 7 x y^{2} x^{2} y - 3 x + 7 x y^{2}$

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23.

ⓐ $x y^{2} - 5 x - 5 y^{2} x y^{2} - 5 x - 5 y^{2}$
ⓑ $19 y + 5 z 19 y + 5 z$

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25.

$x - 3 y + a^{2} b - 4 a - 5 a b^{2} x - 3 y + a^{2} b - 4 a - 5 a b^{2}$

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27.

$− u^{2} + 4 v^{2} + 12a + 8 b − u^{2} + 4 v^{2} +$

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29.

$− 4a - 3 b − 4a - 3 b$

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31.

$−17 x^{6} −17 x^{6}$

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33.

$11 y^{2} + 4 y + 11 11 y^{2} + 4 y + 11$

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35.

$−3 x^{2} + 17 x - 1 −3 x^{2} + 17 x - 1$

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37.

$11 x^{2} - 5 x + 5 11 x^{2} - 5 x + 5$

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39.

$6 a^{2} - 4 a - 1 6 a^{2} - 4 a - 1$

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41.

$2 m^{2} - 7 m + 4 2 m^{2} - 7 m + 4$

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43.

$11 a + 3 11 a + 3$

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45.

$12 s^{2} - 16 s + 9 12 s^{2} - 16 s + 9$

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47.

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

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49.

$w^{2} + 3 w + 4 w^{2} + 3 w + 4$

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51.

$11 w - 66 11 w - 66$

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53.

$10 x^{2} - 7 x y + 6 y^{2} 10 x^{2} - 7 x y + 6 y^{2}$

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55.

$10 m^{2} + 3 m n - 8 n^{2} 10 m^{2} + 3 m n - 8 n^{2}$

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57.

$−3 a b + 3 b^{2} −3 a b + 3 b^{2}$

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59.

$p^{3} - 6 p^{2} q + p q^{2} + 4 q^{3} p^{3} - 6 p^{2} q + p q^{2} + 4 q^{3}$

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61.

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

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63.

ⓐ 187 ⓑ 40 ⓒ 2

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65.

ⓐ $−104 −104$ ⓑ 4 ⓒ 40

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67.

The height is 11 feet.

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69.

The revenue is $10,800.

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71.

The cost is $456.

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73.

ⓐ $(f + g) (x) = 7 x^{2} + 4 x + 4 (f + g) (x) = 7 x^{2} + 4 x + 4$ ⓑ $(f + g) (2) = 40 (f + g) (2) = 40$
ⓒ $(f - g) (x) = −3 x^{2} - 12 x - 2 (f - g) (x) = −3 x^{2} - 12 x - 2$
ⓓ $(f - g) (−3) = 7 (f - g) (−3) = 7$

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75.

ⓐ $(f + g) (x) = 6 x^{3} - x^{2} - 9 x + 3 (f + g) (x) = 6 x^{3} - x^{2} - 9 x + 3$
ⓑ $(f + g) (2) = 29 (f + g) (2) = 29$
ⓒ $(f - g) (x) = − x^{2} + 5 x + 3 (f - g) (x) = − x^{2} + 5 x + 3$
ⓓ $(f - g) (−3) = −21 (f - g) (−3) = −21$

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77.

Answers will vary.

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79.

Answers will vary.

Section 5.2 Exercises

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81.

ⓐ $d^{9} d^{9}$ ⓑ $4^{14 x} 4^{14 x}$ ⓒ $8 y^{4} 8 y^{4}$ ⓓ $w^{6} w^{6}$

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83.

ⓐ $n^{31} n^{31}$ ⓑ $3^{x + 6} 3^{x + 6}$ ⓒ $56 w^{6} 56 w^{6}$
ⓓ $a^{16} a^{16}$

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85.

$m^{x + 3} m^{x + 3}$

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87.

$y^{a + b} y^{a + b}$

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89.

ⓐ $x^{15} x^{15}$ ⓑ $5^{9} 5^{9}$ ⓒ $\frac{1}{q^{18}} \frac{1}{q^{18}}$ ⓓ $\frac{1}{10} \frac{1}{10}$

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91.

ⓐ $p^{14} p^{14}$ ⓑ $4^{12} 4^{12}$ ⓒ $\frac{1}{b^{8}} \frac{1}{b^{8}}$ ⓓ $\frac{1}{4^{5}} \frac{1}{4^{5}}$

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93.

ⓐ 1 ⓑ 1

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95.

ⓐ $−1 −1$ ⓑ $−1 −1$

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97.

ⓐ $\frac{1}{a^{2}} \frac{1}{a^{2}}$ ⓑ $\frac{1}{1000} \frac{1}{1000}$ ⓒ $c^{5} c^{5}$ ⓓ $99$

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99.

ⓐ $\frac{1}{r^{3}} \frac{1}{r^{3}}$ ⓑ $\frac{1}{100,000} \frac{1}{100,000}$ ⓒ $q^{10} q^{10}$
ⓓ $1,000 1,000$

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101.

ⓐ $\frac{64}{25} \frac{64}{25}$ ⓑ $\frac{a^{2}}{b^{2}} \frac{a^{2}}{b^{2}}$

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103.

ⓐ $\frac{729}{64} \frac{729}{64}$ ⓑ $- \frac{v^{5}}{u^{5}} - \frac{v^{5}}{u^{5}}$

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105.

ⓐ $\frac{1}{25} \frac{1}{25}$ ⓑ $\frac{1}{25} \frac{1}{25}$ ⓒ $2525$ ⓓ $−25 −25$

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107.

ⓐ $\frac{3}{5} \frac{3}{5}$ ⓑ $\frac{1}{15} \frac{1}{15}$

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109.

ⓐ $\frac{1}{b^{4}} \frac{1}{b^{4}}$ ⓑ $\frac{w^{2}}{x^{9}} \frac{w^{2}}{x^{9}}$ ⓒ $−12 c d^{4} −12 c d^{4}$

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111.

ⓐ 1 ⓑ $\frac{1}{u^{4} v^{5}} \frac{1}{u^{4} v^{5}}$ ⓒ $−36 \frac{r^{2}}{j^{5}} −36 \frac{r^{2}}{j^{5}}$

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113.

$\frac{1}{p} \frac{1}{p}$

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115.

ⓐ $m^{8} m^{8}$ ⓑ $10^{18} 10^{18}$ ⓒ $\frac{1}{x^{12}} \frac{1}{x^{12}}$

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117.

ⓐ $y^{3 x} y^{3 x}$ ⓑ $5^{x y} 5^{x y}$ ⓒ $\frac{1}{q^{48}} \frac{1}{q^{48}}$

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119.

ⓐ $9 x^{2} y^{2} 9 x^{2} y^{2}$ ⓑ 1 ⓒ $\frac{1}{25 x^{4}} \frac{1}{25 x^{4}}$
ⓓ $\frac{16}{y^{6}} \frac{16}{y^{6}}$

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121.

ⓐ $−125 a^{3} b^{3} −125 a^{3} b^{3}$ ⓑ 1 ⓒ $\frac{1}{36 x^{6}} \frac{1}{36 x^{6}}$ ⓓ $\frac{9}{y^{8}} \frac{9}{y^{8}}$

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123.

ⓐ $\frac{p^{5}}{32} \frac{p^{5}}{32}$ ⓑ $\frac{y^{6}}{x^{6}} \frac{y^{6}}{x^{6}}$ ⓒ $\frac{8 x^{3} y^{6}}{z^{3}} \frac{8 x^{3} y^{6}}{z^{3}}$
ⓓ $\frac{16}{p^{6} q^{4}} \frac{16}{p^{6} q^{4}}$

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125.

ⓐ $\frac{a^{4}}{81 b^{4}} \frac{a^{4}}{81 b^{4}}$ ⓑ $\frac{16 m^{2}}{25} \frac{16 m^{2}}{25}$ ⓒ $\frac{a^{4} c^{4}}{9 b^{6}} \frac{a^{4} c^{4}}{9 b^{6}}$ ⓓ $\frac{q^{8} r^{8}}{p^{2}} \frac{q^{8} r^{8}}{p^{2}}$

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127.

ⓐ $1125 t^{8} 1125 t^{8}$ ⓑ $\frac{1}{t^{19}} \frac{1}{t^{19}}$ ⓒ $\frac{y^{4}}{3 x^{2}} \frac{y^{4}}{3 x^{2}}$

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129.

ⓐ $16 m^{8} n^{22} 16 m^{8} n^{22}$ ⓑ $\frac{4}{p^{6}} \frac{4}{p^{6}}$

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131.

ⓐ $\frac{7}{n} \frac{7}{n}$ ⓑ $\frac{1}{7 n} \frac{1}{7 n}$ ⓒ $- \frac{1}{7 n} - \frac{1}{7 n}$

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133.

ⓐ $\frac{1}{9 p^{2}} \frac{1}{9 p^{2}}$ ⓑ $\frac{3}{p^{2}} \frac{3}{p^{2}}$ ⓒ $\frac{−3}{p^{2}} \frac{−3}{p^{2}}$

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135.

$x^{14} x^{14}$

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137.

$x^{30} x^{30}$

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139.

$8 m^{18} 8 m^{18}$

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141.

$1,000 x^{6} y^{3} 1,000 x^{6} y^{3}$

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143.

$16 a^{12} b^{8} 16 a^{12} b^{8}$

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145.

$\frac{8}{27} x^{6} y^{3} \frac{8}{27} x^{6} y^{3}$

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147.

$1,024 a^{10} 1,024 a^{10}$

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149.

$25,000 p^{24} 25,000 p^{24}$

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151.

$x^{18} y^{18} x^{18} y^{18}$

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153.

$144 m^{8} n^{22} 144 m^{8} n^{22}$

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155.

ⓐ $45 x^{3} 45 x^{3}$ ⓑ $48 y^{4} 48 y^{4}$

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157.

ⓐ $\frac{1}{2 r^{4}} \frac{1}{2 r^{4}}$ ⓑ $\frac{1}{3} x^{11} \frac{1}{3} x^{11}$

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159.

$\frac{1}{j^{3}} \frac{1}{j^{3}}$

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161.

$- \frac{4000}{n^{12}} - \frac{4000}{n^{12}}$

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163.

ⓐ $34 \times 10^{4} 34$ ⓑ $41 \times 10^{−3} 41$

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165.

ⓐ $1.29 \times 10^{6} 1.29$
ⓑ $103 \times 10^{−8} 103$

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167.

ⓐ $−830 −830$ ⓑ 0.038

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169.

ⓐ 16,000,000,000
ⓑ 0.00000843

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171.

ⓐ 0.02 ⓑ 500,000,000

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173.

ⓐ 0.0000056 ⓑ 20,000,000

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175.

Answers will vary.

Show solution

177.

Answers will vary.

Section 5.3 Exercises

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179.

ⓐ $30 x^{8} 30 x^{8}$ ⓑ $15 x^{8} y^{2} 15 x^{8} y^{2}$

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181.

ⓐ $72 c^{5} 72 c^{5}$ ⓑ $\frac{1}{3} m^{5} n^{5} \frac{1}{3} m^{5} n^{5}$

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183.

ⓐ $−5 t^{3} - 15 t^{2} + 90 t −5 t^{3} - 15 t^{2} + 90 t$
ⓑ $9 s r^{5} - 27 s^{2} r^{4} + 45 s^{3} r^{3} 9 s r^{5} - 27 s^{2} r^{4} + 45 s^{3} r^{3}$

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185.

ⓐ $−5 m^{3} - 15 m^{2} + 90 m −5 m^{3} - 15 m^{2} + 90 m$
ⓑ $−21 x^{4} y^{2} - 30 x^{3} y^{3} + 3 x^{2} y^{4} −21 x^{4} y^{2} - 30 x^{3} y^{3} + 3 x^{2} y^{4}$

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187.

$y^{2} + 12 y + 27 y^{2} + 12 y + 27$

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189.

$21 q^{2} - 44 q - 32 21 q^{2} - 44 q - 32$

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191.

$y^{2} - 8 y + 12 y^{2} - 8 y + 12$

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193.

$6 p^{2} + 11 p + 5 6 p^{2} + 11 p + 5$

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195.

$m^{2} + 7 m - 44 m^{2} + 7 m - 44$

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197.

$33 r^{2} - 85 r - 8 33 r^{2} - 85 r - 8$

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199.

$y^{3} + 3 y^{2} - 4 y - 12 y^{3} + 3 y^{2} - 4 y - 12$

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201.

$6 x^{2} y^{2} + 13 x y + 6 6 x^{2} y^{2} + 13 x y + 6$

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203.

$y^{4} - 11 y^{2} + 28 y^{4} - 11 y^{2} + 28$

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205.

$9 r^{2} s^{2} - 33 r s + 28 9 r^{2} s^{2} - 33 r s + 28$

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207.

$u^{3} + 7 u^{2} + 14 u + 8 u^{3} + 7 u^{2} + 14 u + 8$

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209.

$3 a^{3} + 31 a^{2} + 5 a - 50 3 a^{3} + 31 a^{2} + 5 a - 50$

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211.

$6 a^{4} - 13 a^{3} + 15 a^{2} + 35 a - 50 6 a^{4} - 13 a^{3} + 15 a^{2} + 35 a - 50$

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213.

$p^{3} - 10 p^{2} + 33 p - 36 p^{3} - 10 p^{2} + 33 p - 36$

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215.

$6 r^{3} - 41 r^{2} - 61 r - 9 6 r^{3} - 41 r^{2} - 61 r - 9$

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217.

$q^{2} + 24 q + 144 q^{2} + 24 q + 144$

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219.

$4 y^{2} - 12 y z + 9 z^{2} 4 y^{2} - 12 y z + 9 z^{2}$

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221.

$x^{2} + \frac{4}{3} x + \frac{4}{9} x^{2} + \frac{4}{3} x + \frac{4}{9}$

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223.

$\frac{1}{64} x^{2} - \frac{1}{36} x y + \frac{1}{81} y^{2} \frac{1}{64} x^{2} - \frac{1}{36} x y + \frac{1}{81} y^{2}$

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225.

$25 u^{4} + 90 u^{2} + 81 25 u^{4} + 90 u^{2} + 81$

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227.

$64 p^{6} - 48 p^{3} + 9 64 p^{6} - 48 p^{3} + 9$

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229.

$64 j^{2} - 16 64 j^{2} - 16$

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231.

$81 c^{2} - 25 81 c^{2} - 25$

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233.

$49 w^{2} - 100 x^{2} 49 w^{2} - 100 x^{2}$

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235.

$p^{2} - \frac{16}{25} q^{2} p^{2} - \frac{16}{25} q^{2}$

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237.

$x^{2} y^{2} - 81 x^{2} y^{2} - 81$

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239.

$225 m^{4} - 64 n^{8} 225 m^{4} - 64 n^{8}$

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241.

$t^{2} - 18 t + 81 t^{2} - 18 t + 81$

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243.

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

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245.

$9 p^{2} - 64 9 p^{2} - 64$

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247.

$k^{2} - 12 k + 36 k^{2} - 12 k + 36$

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249.

$8 x^{3} - x^{2} y^{2} + 64 x y - 8 y^{3} 8 x^{3} - x^{2} y^{2} + 64 x y - 8 y^{3}$

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251.

$y^{8} + 4 y^{4} z + 4 z^{2} y^{8} + 4 y^{4} z + 4 z^{2}$

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253.

$m^{6} - 16 m^{3} n + 64 n^{2} m^{6} - 16 m^{3} n + 64 n^{2}$

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255.

$r^{5} + r^{2} s^{2} - r^{3} s^{3} - s^{5} r^{5} + r^{2} s^{2} - r^{3} s^{3} - s^{5}$

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257.

$18 p - 9 18 p - 9$

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259.

$−10 j - 15 −10 j - 15$

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261.

$9 d^{7} 9 d^{7}$

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263.

$42 m^{5} n^{8} 42 m^{5} n^{8}$

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265.

$5 q^{5} - 10 q^{4} + 30 q^{3} 5 q^{5} - 10 q^{4} + 30 q^{3}$

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267.

$y^{3} - y^{2} - 2 y y^{3} - y^{2} - 2 y$

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269.

$6 k^{3} + 11 k^{2} - 26 k + 4 6 k^{3} + 11 k^{2} - 26 k + 4$

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271.

$121 - b^{2} 121 - b^{2}$

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273.

$4 x^{4} - 9 y^{8} 4 x^{4} - 9 y^{8}$

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275.

$9 d^{2} + 6 d + 1 9 d^{2} + 6 d + 1$

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277.

$9 z^{2} + \frac{6}{5} z + \frac{1}{25} 9 z^{2} + \frac{6}{5} z + \frac{1}{25}$

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279.

ⓐ $(f \cdot g) (x) = 4 x^{3} - x^{2} - 8 x + 5 (f \cdot g) (x) = 4 x^{3} - x^{2} - 8 x + 5$
ⓑ $(f \cdot g) (−2) = −15 (f \cdot g) (−2) = −15$

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281.

ⓐ $(f \cdot g) (x) = 49 x^{2} - 64 (f \cdot g) (x) = 49 x^{2} - 64$
ⓑ $(f \cdot g) (−2) = 187 (f \cdot g) (−2) = 187$

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283.

ⓐ $(f \cdot g) (x) = x^{4} + 6 x^{3} + 9 x^{2} + 10 x - 12 (f \cdot g) (x) = x^{4} + 6 x^{3} + 9 x^{2} + 10 x - 12$ ⓑ $(f \cdot g) (1) = 14 (f \cdot g) (1) = 14$

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285.

Answers will vary.

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287.

Answers will vary.

Section 5.4 Exercises

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289.

$\frac{2 m^{3}}{3 n^{5}} \frac{2 m^{3}}{3 n^{5}}$

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291.

$\frac{−3 y^{3}}{4 x^{3}} \frac{−3 y^{3}}{4 x^{3}}$

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293.

$\frac{−3 q^{5}}{p^{5}} \frac{−3 q^{5}}{p^{5}}$

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295.

$\frac{5 v^{4}}{u^{2}} \frac{5 v^{4}}{u^{2}}$

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297.

$4 x^{2} + 3 x 4 x^{2} + 3 x$

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299.

$−6 y^{2} + 3 y −6 y^{2} + 3 y$

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301.

$6 r^{3} + 11 r^{2} s - 8 r s^{3} 6 r^{3} + 11 r^{2} s - 8 r s^{3}$

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303.

$−5 y - 3 + \frac{1}{4 y} −5 y - 3 + \frac{1}{4 y}$

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305.

$a - 7 a - 7$

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307.

$4 x + 3 4 x + 3$

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309.

$p + 3 - \frac{8}{p + 8} p + 3 - \frac{8}{p + 8}$

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311.

$\frac{2 n^{2} - 6 n + 8 + 4}{n + 3} \frac{2 n^{2} - 6 n + 8 + 4}{n + 3}$

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313.

$m^{2} - 10 m + 100 m^{2} - 10 m + 100$

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315.

$25 y^{2} + 20 x + 16 25 y^{2} + 20 x + 16$

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317.

$x^{2} - 5 x + 6; 0 x^{2} - 5 x + 6;$

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319.

$2 x^{2} - 3 x + 4; 4 2 x^{2} - 3 x + 4;$

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321.

$x^{3} - 2 x^{2} + 5 x - 4; − 2 x^{3} - 2 x^{2} + 5 x - 4;$

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323.

$3 x^{3} - 2 x^{2} - 4 x - 2; 0 3 x^{3} - 2 x^{2} - 4 x - 2;$

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325.

ⓐ $(\frac{f}{g}) (x) = x - 6 (\frac{f}{g}) (x) = x - 6$
ⓑ $(\frac{f}{g}) (−5) = −11 (\frac{f}{g}) (−5) = −11$

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327.

ⓐ $(\frac{f}{g}) (x) = x^{2} + 5 x - 4 (\frac{f}{g}) (x) = x^{2} + 5 x - 4$
ⓑ $(\frac{f}{g}) (0) = −4 (\frac{f}{g}) (0) = −4$

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329.

ⓐ $(f \cdot g) (x) = x^{4} + 6 x^{3} + 9 x^{2} + 10 x - 12; (f \cdot g) (x) = x^{4} + 6 x^{3} + 9 x^{2} + 10 x - 12;$ ⓑ $(f \cdot g) (1) = 14 (f \cdot g) (1) = 14$

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331.

$−9 −9$

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333.

−6

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335.

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337.

yes

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339.

answer will vary

Show solution

341.

Answers will vary.

Review Exercises

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343.

binomial

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345.

other polynomial

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347.

$−13 y^{3} −13 y^{3}$

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349.

$6 m^{2} + 19 m - 4 6 m^{2} + 19 m - 4$

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351.

$5 u^{2} + 4 u + 7 5 u^{2} + 4 u + 7$

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353.

$2 x^{2} - 2 x + 23 2 x^{2} - 2 x + 23$

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355.

$−7 b - 3 a −7 b - 3 a$

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357.

$8 a^{2} - 5 a - 2 8 a^{2} - 5 a - 2$

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359.

$−3 m + 3 −3 m + 3$

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361.

$3 y^{2} - 8 y - 14 3 y^{2} - 8 y - 14$

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363.

$x^{3} + 2 x^{2} y - 4 x y^{2} x^{3} + 2 x^{2} y - 4 x y^{2}$

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365.

ⓐ 165 ⓑ 39 ⓒ 5

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367.

The height is $64 feet 64$ .

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369.

ⓐ $(f + g) (x) = 4 x^{2} - 5 x - 2 (f + g) (x) = 4 x^{2} - 5 x - 2$ ⓑ $(f + g) (3) = 19 (f + g) (3) = 19$
ⓒ $(f - g) (x) = −3 x - 12 (f - g) (x) = −3 x - 12$
ⓓ $(f - g) (−2) = −6 (f - g) (−2) = −6$

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371.

$p^{13} p^{13}$

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373.

$a^{6} a^{6}$

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375.

$y^{a + b} y^{a + b}$

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377.

$a^{5} a^{5}$

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379.

$\frac{1}{x^{4}} \frac{1}{x^{4}}$

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381.

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383.

$−3 −3$

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385.

$- \frac{1}{1000} - \frac{1}{1000}$

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387.

$\frac{1}{8 n} \frac{1}{8 n}$

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389.

$\frac{1}{1000} \frac{1}{1000}$

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391.

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393.

$−125 −125$

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395.

$- \frac{1}{125} - \frac{1}{125}$

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397.

$\frac{x^{3}}{27} \frac{x^{3}}{27}$

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399.

$3^{10} 3^{10}$

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401.

$x^{5} x^{5}$

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403.

$\frac{1}{u^{3} v^{5}} \frac{1}{u^{3} v^{5}}$

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405.

$\frac{1}{m^{5}} \frac{1}{m^{5}}$

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407.

$\frac{1}{q^{16}} \frac{1}{q^{16}}$

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409.

$n^{2} n^{2}$

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411.

Show solution

413.

$\frac{9}{y^{8}} \frac{9}{y^{8}}$

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415.

$\frac{81 x^{4} y^{8}}{z^{4}} \frac{81 x^{4} y^{8}}{z^{4}}$

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417.

$27 x^{7} y^{17} 27 x^{7} y^{17}$

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419.

$\frac{3 y^{4}}{4 x^{4}} \frac{3 y^{4}}{4 x^{4}}$

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421.

$5.3 \times 10^{6} 5.3$

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423.

$29,000 29,000$

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425.

$0.00009413 0.00009413$

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427.

$0.00072 0.00072$

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429.

$9,000 9,000$

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431.

$10 c^{10} 10 c^{10}$

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433.

$\frac{m^{7} n^{10}}{9} \frac{m^{7} n^{10}}{9}$

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435.

$a^{4} - 9 a^{3} - 36 a^{2} a^{4} - 9 a^{3} - 36 a^{2}$

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437.

$8 n^{4} - 10 n^{3} 8 n^{4} - 10 n^{3}$

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439.

$y^{2} + 8 y - 48 y^{2} + 8 y - 48$

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441.

$18 p^{2} - 93 p + 110 18 p^{2} - 93 p + 110$

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443.

$k^{2} - 3 k - 54 k^{2} - 3 k - 54$

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445.

$10 y^{2} - 59 y + 63 10 y^{2} - 59 y + 63$

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447.

$x^{2} + x - 72 x^{2} + x - 72$

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449.

$30 a^{2} - 33 a + 3 30 a^{2} - 33 a + 3$

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451.

$15 b^{3} - b^{2} - 47 b + 18 15 b^{3} - b^{2} - 47 b + 18$

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453.

$24 y^{2} - 54 y^{2} + 32 y - 5 24 y^{2} - 54 y^{2} + 32 y - 5$

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455.

$x^{2} + \frac{3}{2} x + \frac{9}{16} x^{2} + \frac{3}{2} x + \frac{9}{16}$

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457.

$25 p^{2} + 70 p q + 49 q^{2} 25 p^{2} + 70 p q + 49 q^{2}$

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459.

$36 x^{2} - y^{2} 36 x^{2} - y^{2}$

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461.

$144 x^{6} - 49 y^{4} 144 x^{6} - 49 y^{4}$

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463.

$9 p^{9} 9 p^{9}$

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465.

$- \frac{3}{y^{4}} - \frac{3}{y^{4}}$

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467.

$\frac{4 a^{5}}{b^{2}} \frac{4 a^{5}}{b^{2}}$

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469.

$\frac{4 m^{9}}{n^{4}} \frac{4 m^{9}}{n^{4}}$

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471.

$−9 y^{2} + 4 y −9 y^{2} + 4 y$

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473.

$−3 x - 1 + \frac{3}{4 x} −3 x - 1 + \frac{3}{4 x}$

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475.

$y - 3 + \frac{33}{q + 6} y - 3 + \frac{33}{q + 6}$

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477.

$a^{2} + a + 1 a^{2} + a + 1$

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479.

$2 x^{2} - 5 x - 4; 0 2 x^{2} - 5 x - 4;$

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481.

ⓐ $(\frac{f}{g}) (x) = x - 6 (\frac{f}{g}) (x) = x - 6$
ⓑ $(\frac{f}{g}) (−2) = −8 (\frac{f}{g}) (−2) = −8$

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483.

$−9 −9$

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485.

Practice Test

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487.

ⓐ trinomial ⓑ 4

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489.

$6 x^{2} - 3 x + 11 6 x^{2} - 3 x + 11$

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491.

$x x$

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493.

$11$

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495.

$\frac{1}{8 y^{3}} \frac{1}{8 y^{3}}$

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497.

$x^{9} x^{9}$

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499.

$\frac{4 r}{s^{6}} \frac{4 r}{s^{6}}$

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501.

$−48 x^{5} y^{9} −48 x^{5} y^{9}$

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503.

$21 m^{2} - 19 m - 6 21 m^{2} - 19 m - 6$

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505.

$16 x^{2} - 24 x + 9 16 x^{2} - 24 x + 9$

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507.

$3 y^{2} - 7 x 3 y^{2} - 7 x$

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509.

yes

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511.

$4.4 \times 10^{3} 4.4$

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513.

ⓐ $3636$ ⓑ $2121$

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515.

ⓐ $(\frac{f}{g}) (x) = 3 x + 4 (\frac{f}{g}) (x) = 3 x + 4$
ⓑ $(\frac{f}{g}) (3) = 13 (\frac{f}{g}) (3) = 13$