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Intermediate Algebra

Chapter 5

Intermediate AlgebraChapter 5
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Be Prepared

Try It

5.1

monomial, 0
polynomial, 3 trinomial, 3
binomial, 2 monomial, 10

5.2

binomial, 3 trinomial, 3 monomial, 0 polynomial, 4 monomial, 7

5.3

21q221q2 13mn313mn3

5.4

−7c2−7c2 −10y2z3−10y2z3

5.5

5y2+3z25y2+3z2
m2n28m2+4n2m2n28m2+4n2

5.6

−4m2+n2−4m2+n2
pq26p5q2pq26p5q2

5.7

8x211x+88x211x+8

5.8

17y2+14y+117y2+14y+1

5.9

x2+3x5x2+3x5

5.10

6b2+36b2+3

5.11

−5ab+7b2−5ab+7b2

5.12

7mn+4n27mn+4n2

5.13

5x25xy+5y25x25xy+5y2

5.14

7x26xy2y27x26xy2y2

5.15

x3+y3x3+y3

5.16

p33p2q+q3p33p2q+q3

5.17

18 50 −15−15

5.18

20 2 −4−4

5.19

The height is 150150 feet.

5.20

The height is 31 feet.

5.21

(f+g)(x)=3x26x3(f+g)(x)=3x26x3 (f+g)(3)=6(f+g)(3)=6
(fg)(x)=x22x+9(fg)(x)=x22x+9
(fg)(−2)=17(fg)(−2)=17

5.22

(f+g)(x)=6x2x+7(f+g)(x)=6x2x+7 (f+g)(3)=58(f+g)(3)=58
(fg)(x)=4x27x9(fg)(x)=4x27x9
(fg)(−2)=21(fg)(−2)=21

5.23

b17b17 43x43x 12p612p6
x18x18

5.24

x16x16 10x+110x+1 12z812z8
b19b19

5.25

x5x5 6969 1x41x4
1121511215

5.26

y6y6 108108 1m81m8
19111911

5.27

1 1

5.28

1 1

5.29

1z31z3 11071107 p8p8 6464

5.30

1n21n2 110,000110,000 q7q7
1616

5.31

81168116 n2m2n2m2

5.32

1252712527 b4a4b4a4

5.33

1z91z9 1p3q31p3q3 12v5u12v5u

5.34

1c151c15 1r2s81r2s8 30d3c830d3c8

5.35

b35b35 512512 a48a48

5.36

z54z54 349349 q29q29

5.37

32w5x532w5x5 1 116b12116b12
64a864a8

5.38

−27y3−27y3 1 116x8116x8
8c128c12

5.39

p410000p410000 n7m7n7m7
81a4b12c881a4b12c8 27x6y927x6y9

5.40

−8q3−8q3 x4w4x4w4 x2y69z4x2y69z4
8n6m68n6m6

5.41

81c24d3081c24d30 1a181a18
y15y15

5.42

256a22b24256a22b24 1p391p39
2x3y102x3y10

5.43

9.6×1049.6×104 7.8×10−37.8×10−3

5.44

4.83×1044.83×104
1.29×10−21.29×10−2

5.45

1,300 −0.00012−0.00012

5.46

−950,000−950,000 0.075

5.47

−0.006−0.006 20,000

5.48

−0.009−0.009 400,000

5.49

−35y11−35y11 6a5b66a5b6

5.50

54b954b9 8r11s88r11s8

5.51

−15y324y2+21y−15y324y2+21y
12x4y220x3y3+12x2y412x4y220x3y3+12x2y4

5.52

8x424x3+20x28x424x3+20x2
−18a5b+12a4b236a3b3−18a5b+12a4b236a3b3

5.53

x2+17x+72x2+17x+72
15c2+14c815c2+14c8

5.54

20x2+51x+2720x2+51x+27
30y23y630y23y6

5.55

x22x35x22x35
15x2+29x1415x2+29x14

5.56

b2+3b18b2+3b18
16y220y5016y220y50

5.57

x38x2+6x48x38x2+6x48
8a2b2+12ab208a2b2+12ab20

5.58

y39y2+7y63y39y2+7y63
8x2y2+2xy158x2y2+2xy15

5.59

15m251m+4215m251m+42

5.60

42b253b+1542b253b+15

5.61

y38y2+17y6y38y2+17y6
y38y2+17y6y38y2+17y6

5.62

2x3+5x27x+202x3+5x27x+20
y38y2+17y6y38y2+17y6

5.63

x2+18x+81x2+18x+81
4c24cd+d24c24cd+d2

5.64

y2+22y+121y2+22y+121
16x240xy+25y216x240xy+25y2

5.65

36x22536x225
16p249q216p249q2

5.66

4x2494x249 9x2y29x2y2

5.67

FOIL; 18b2+77b1818b2+77b18
Binomial Squares; 81p272p+1681p272p+16
Binomial Squares; 49y2+14y+149y2+14y+1
Product of Conjugates; 16r2916r29

5.68

Binomial Squares; 36x2+84x+4936x2+84x+49 Product of Conjugates; 9x2169x216 FOIL; 10x229x+1010x229x+10 Binomial Squares; 36n212n+136n212n+1

5.69

(f·g)(x)=x37x2+13x15(f·g)(x)=x37x2+13x15
(f·g)(2)=−9(f·g)(2)=−9

5.70

(f·g)(x)=x3+x252x28(f·g)(x)=x3+x252x28
(f·g)(2)=−120(f·g)(2)=−120

5.71

9a5b9a5b

5.72

−9dc4−9dc4

5.73

4y27x44y27x4

5.74

58m5n358m5n3

5.75

−4a+2b−4a+2b

5.76

8a5b+6a3b28a5b+6a3b2

5.77

y+7y+7

5.78

m+5m+5

5.79

x33x2+2x+1+3x+3x33x2+2x+1+3x+3

5.80

x33x22x13x+3x33x22x13x+3

5.81

x2+4x+16x2+4x+16

5.82

25x2+10x+425x2+10x+4

5.83

3x2+4x2;23x2+4x2;2

5.84

4x23x+1;14x23x+1;1

5.85

x34x2+5;0x34x2+5;0

5.86

x33x2+2;0x33x2+2;0

5.87

(fg)(x)=x8(fg)(x)=x8
(fg)(−3)=−11(fg)(−3)=−11

5.88

(fg)(x)=x9(fg)(x)=x9
(fg)(−5)=−14(fg)(−5)=−14

5.89

−1−1

5.90

66

5.91

yes

5.92

yes

Section 5.1 Exercises

1.

trinomial, 5 polynomial, 3 binomial, 2 monomial, 0
binomial, 1

3.

binomial trinomial
polynomial trinomial
monomial

5.

trinomial, 2trinomial, 2 monomial, 4monomial, 4 binomial, 1binomial, 1 quadrinomial, 3quadrinomial, 3
monomial, 0monomial, 0

7.

binomial, 1binomial, 1 trinomial, 2trinomial, 2 pentanomial, 3pentanomial, 3 binomial, 3binomial, 3
monomial, 0monomial, 0

9.

12x212x2 5a5a

11.

6w6w 19x2y19x2y

13.

12x25a12x25a

15.

6w+19x2y6w+19x2y

17.

−22b−22b 16xy16xy

19.

−10a+5b−10a+5b
pq24p3q2pq24p3q2

21.

−4a2+b2−4a2+b2
x2y3x+7xy2x2y3x+7xy2

23.

xy25x5y2xy25x5y2
19y+5z19y+5z

25.

x3y+a2b4a5ab2x3y+a2b4a5ab2

27.

u2+4v2+12a+8bu2+4v2+12a+8b

29.

4a3b4a3b

31.

−17x6−17x6

33.

11y2+4y+1111y2+4y+11

35.

−3x2+17x1−3x2+17x1

37.

11x25x+511x25x+5

39.

6a24a16a24a1

41.

2m27m+42m27m+4

43.

11a+311a+3

45.

12s216s+912s216s+9

47.

3x2x+43x2x+4

49.

w2+3w+4w2+3w+4

51.

11w6611w66

53.

10x27xy+6y210x27xy+6y2

55.

10m2+3mn8n210m2+3mn8n2

57.

−3ab+3b2−3ab+3b2

59.

p36p2q+pq2+4q3p36p2q+pq2+4q3

61.

x3+2x2y5xy2+y3x3+2x2y5xy2+y3

63.

187 40 2

65.

−104−104 4 40

67.

The height is 11 feet.

69.

The revenue is $10,800.

71.

The cost is $456.

73.

(f+g)(x)=7x2+4x+4(f+g)(x)=7x2+4x+4 (f+g)(2)=40(f+g)(2)=40
(fg)(x)=−3x212x2(fg)(x)=−3x212x2
(fg)(−3)=7(fg)(−3)=7

75.


 (f+g)(x)=6x3x29x+3(f+g)(x)=6x3x29x+3
(f+g)(2)=29(f+g)(2)=29
(fg)(x)=x2+5x+3(fg)(x)=x2+5x+3
(fg)(−3)=−21(fg)(−3)=−21

77.

Answers will vary.

79.

Answers will vary.

Section 5.2 Exercises

81.

d9d9 414x414x 8y48y4 w6w6

83.

n31n31 3x+63x+6 56w656w6
a16a16

85.

mx+3mx+3

87.

ya+bya+b

89.

x15x15 5959 1q181q18 110110

91.

p14p14 412412 1b81b8 145145

93.

1 1

95.

−1−1 −1−1

97.

1a21a2 1100011000 c5c5 99

99.

1r31r3 1100,0001100,000 q10q10
1,0001,000

101.

64256425 a2b2a2b2

103.

7296472964 v5u5v5u5

105.

125125 125125 2525 −25−25

107.

3535 115115

109.

1b41b4 w2x9w2x9 −12cd4−12cd4

111.

1 1u4v51u4v5 −36r2j5−36r2j5

113.

1p1p

115.

m8m8 10181018 1x121x12

117.

y3xy3x 5xy5xy 1q481q48

119.

9x2y29x2y2 1 125x4125x4
16y616y6

121.

−125a3b3−125a3b3 1 136x6136x6 9y89y8

123.

p532p532 y6x6y6x6 8x3y6z38x3y6z3
16p6q416p6q4

125.

a481b4a481b4 16m22516m225 a4c49b6a4c49b6 q8r8p2q8r8p2

127.

1125t81125t8 1t191t19 y43x2y43x2

129.

16m8n2216m8n22 4p64p6

131.

7n7n 17n17n 17n17n

133.

19p219p2 3p23p2 −3p2−3p2

135.

x14x14

137.

x30x30

139.

8m188m18

141.

1,000x6y31,000x6y3

143.

16a12b816a12b8

145.

827x6y3827x6y3

147.

1,024a101,024a10

149.

25,000p2425,000p24

151.

x18y18x18y18

153.

144m8n22144m8n22

155.

45x345x3 48y448y4

157.

12r412r4 13x1113x11

159.

1j31j3

161.

4000n124000n12

163.

34×10434×104 41×10−341×10−3

165.

1.29×1061.29×106
103×10−8103×10−8

167.

−830−830 0.038

169.

16,000,000,000
0.00000843

171.

0.02 500,000,000

173.

0.0000056 20,000,000

175.

Answers will vary.

177.

Answers will vary.

Section 5.3 Exercises

179.

30x830x8 15x8y215x8y2

181.

72c572c5 13m5n513m5n5

183.

−5t315t2+90t−5t315t2+90t
9sr527s2r4+45s3r39sr527s2r4+45s3r3

185.

−5m315m2+90m−5m315m2+90m
−21x4y230x3y3+3x2y4−21x4y230x3y3+3x2y4

187.

y2+12y+27y2+12y+27

189.

21q244q3221q244q32

191.

y28y+12y28y+12

193.

6p2+11p+56p2+11p+5

195.

m2+7m44m2+7m44

197.

33r285r833r285r8

199.

y3+3y24y12y3+3y24y12

201.

6x2y2+13xy+66x2y2+13xy+6

203.

y411y2+28y411y2+28

205.

9r2s233rs+289r2s233rs+28

207.

u3+7u2+14u+8u3+7u2+14u+8

209.

3a3+31a2+5a503a3+31a2+5a50

211.

6a413a3+15a2+35a506a413a3+15a2+35a50

213.

p310p2+33p36p310p2+33p36

215.

6r341r261r96r341r261r9

217.

q2+24q+144q2+24q+144

219.

4y212yz+9z24y212yz+9z2

221.

x2+43x+49x2+43x+49

223.

164x2136xy+181y2164x2136xy+181y2

225.

25u4+90u2+8125u4+90u2+81

227.

64p648p3+964p648p3+9

229.

64j216 64j216

231.

81c22581c225

233.

49w2100x249w2100x2

235.

p21625q2p21625q2

237.

x2y281 x2y281

239.

225m464n8225m464n8

241.

t218t+81t218t+81

243.

2x23xy2y22x23xy2y2

245.

9p2649p264

247.

k212k+36k212k+36

249.

8x3x2y2+64xy8y38x3x2y2+64xy8y3

251.

y8+4y4z+4z2y8+4y4z+4z2

253.

m616m3n+64n2m616m3n+64n2

255.

r5+r2s2r3s3s5r5+r2s2r3s3s5

257.

18p918p9

259.

−10j15−10j15

261.

9d79d7

263.

42m5n842m5n8

265.

5q510q4+30q35q510q4+30q3

267.

y3y22yy3y22y

269.

6k3+11k226k+46k3+11k226k+4

271.

121b2121b2

273.

4x49y84x49y8

275.

9d2+6d+19d2+6d+1

277.

9z2+65z+1259z2+65z+125

279.


(f·g)(x)=4x3x28x+5(f·g)(x)=4x3x28x+5
(f·g)(−2)=−15(f·g)(−2)=−15

281.

(f·g)(x)=49x264(f·g)(x)=49x264
(f·g)(−2)=187(f·g)(−2)=187

283.


 (f·g)(x)=x4+6x3+9x2+10x12(f·g)(x)=x4+6x3+9x2+10x12 (f·g)(1)=14(f·g)(1)=14

285.

Answers will vary.

287.

Answers will vary.

Section 5.4 Exercises

289.

2m33n52m33n5

291.

−3y34x3−3y34x3

293.

−3q5p5−3q5p5

295.

5v4u25v4u2

297.

4x2+3x4x2+3x

299.

−6y2+3y−6y2+3y

301.

6r3+11r2s8rs36r3+11r2s8rs3

303.

−5y3+14y−5y3+14y

305.

a7a7

307.

4x+34x+3

309.

p+38p+8p+38p+8

311.

2n26n+8+4n+32n26n+8+4n+3

313.

m210m+100m210m+100

315.

25y2+20x+1625y2+20x+16

317.

x25x+6;0x25x+6;0

319.

2x23x+4;42x23x+4;4

321.

x32x2+5x4;2x32x2+5x4;2

323.

3x32x24x2;03x32x24x2;0

325.

(fg)(x)=x6(fg)(x)=x6
(fg)(−5)=−11(fg)(−5)=−11

327.

(fg)(x)=x2+5x4(fg)(x)=x2+5x4
(fg)(0)=−4(fg)(0)=−4

329.


 (f·g)(x)=x4+6x3+9x2+10x12;(f·g)(x)=x4+6x3+9x2+10x12; (f·g)(1)=14(f·g)(1)=14

331.

−9−9

333.

−6

335.

no

337.

yes

339.

answer will vary

341.

Answers will vary.

Review Exercises

343.

binomial

345.

other polynomial

347.

−13y3−13y3

349.

6m2+19m46m2+19m4

351.

5u2+4u+75u2+4u+7

353.

2x22x+232x22x+23

355.

−7b3a−7b3a

357.

8a25a28a25a2

359.

−3m+3−3m+3

361.

3y28y143y28y14

363.

x3+2x2y4xy2x3+2x2y4xy2

365.

165 39 5

367.

The height is 64feet64feet.

369.

(f+g)(x)=4x25x2(f+g)(x)=4x25x2 (f+g)(3)=19(f+g)(3)=19
(fg)(x)=−3x12(fg)(x)=−3x12
(fg)(−2)=−6(fg)(−2)=−6

371.

p13p13

373.

a6a6

375.

ya+bya+b

377.

a5a5

379.

1x41x4

381.

1

383.

−3−3

385.

1100011000

387.

18n18n

389.

1100011000

391.

36

393.

−125−125

395.

11251125

397.

x327x327

399.

310310

401.

x5x5

403.

1u3v51u3v5

405.

1m51m5

407.

1q161q16

409.

n2n2

411.

1

413.

9y89y8

415.

81x4y8z481x4y8z4

417.

27x7y1727x7y17

419.

3y44x43y44x4

421.

5.3×1065.3×106

423.

29,00029,000

425.

0.000094130.00009413

427.

0.000720.00072

429.

9,0009,000

431.

10c1010c10

433.

m7n109m7n109

435.

a49a336a2a49a336a2

437.

8n410n38n410n3

439.

y2+8y48y2+8y48

441.

18p293p+11018p293p+110

443.

k23k54k23k54

445.

10y259y+6310y259y+63

447.

x2+x72x2+x72

449.

30a233a+330a233a+3

451.

15b3b247b+1815b3b247b+18

453.

24y254y2+32y524y254y2+32y5

455.

x2+32x+916x2+32x+916

457.

25p2+70pq+49q225p2+70pq+49q2

459.

36x2y236x2y2

461.

144x649y4144x649y4

463.

9p99p9

465.

3y43y4

467.

4a5b24a5b2

469.

4m9n44m9n4

471.

−9y2+4y−9y2+4y

473.

−3x1+34x−3x1+34x

475.

y3+33q+6y3+33q+6

477.

a2+a+1a2+a+1

479.

2x25x4;02x25x4;0

481.

(fg)(x)=x6(fg)(x)=x6
(fg)(−2)=−8(fg)(−2)=−8

483.

−9−9

485.

no

Practice Test

487.

trinomial 4

489.

6x23x+116x23x+11

491.

xx

493.

11

495.

18y318y3

497.

x9x9

499.

4rs64rs6

501.

−48x5y9−48x5y9

503.

21m219m621m219m6

505.

16x224x+916x224x+9

507.

3y27x3y27x

509.

yes

511.

4.4×1034.4×103

513.

3636 2121

515.

(fg)(x)=3x+4(fg)(x)=3x+4
(fg)(3)=13(fg)(3)=13

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