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Finite Mathematics And Applied Calculus, 6th Edition Solution Manual

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    Complete Solutions Manual Finite Mathematics & Applied Calculus SIXTH EDITION Stefan Waner Hofstra University Steven R. Costenoble Hofstra University

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    Chapter 0 – Precalculus Review 1 Table of Contents Chapter 1 – Functions and Applications 17 Chapter 2 – The Mathematics of Finance 121 Chapter 3 – Systems of Linear Equations and Matrices 172 Chapter 4 – Matrix Algebra and Applications 266 Chapter 5 – Linear Programming 367 Chapter 6 – Sets and Counting 579 Chapter 7 – Probability 632 Chapter 8 – Random Variables and Statistics 734 Chapter 9 – Nonlinear Functions and Models 827 Chapter 10 – Introduction to the Derivative 883 Chapter 11 – Techniques of Differentiation with Applications 962 Chapter 12 – Further Applications of the Derivative 1048 Chapter 13 – The Integral 1167 Chapter 14 – Further Integration Techniques and Applications of the Integral 1233 Chapter 15 – Functions of Several Variables 1337 Chapter 16 – Trigonometric Models 1406

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    Solutions Section 0.1 Section 0.1 1. 2(4 + (-1))(2 · -4) = 2(3)(-8) = (6)(-8) = -48 2. 3 + ([4 - 2] · 9) = 3 + (2 × 9) = 3 + 18 = 21 3 . 20/(3*4)-1 = 20 12 - 1 = 5 3 - 1 = 2 3 4 . 2-(3*4)/10 = 2 - 12 10 = 2 - 6 5 = 4 5 5. 3 + ([3 + (-5)]) 3 - 2¿2 = 3 + (-2) 3 - 4 = 1 -1 = -1 6. 12 - (1 - 4) 2(5 - 1) · 2 - 1 = 12-(-3) 16 - 1 = 15 15 = 1 7. (2-5*(-1))/1-2*(-1) = 2 - 5 · (-1) 1 - 2 · (-1) = 2+5 1 + 2 = 7 + 2 = 9 8. 2-5*(-1)/(1-2*(-1)) = 2 - 5 · (-1) 1 - 2 · (-1) = 2 - -5 1 + 2 = 2 + 5 3 = 11 3 9. 2 · (-1) 2 / 2 = 2¿(-1) 2 2 = 2¿1 2 = 2 2 = 1 10. 2 + 4 · 3 2 = 2 + 4 × 9 = 2 + 36 = 38 11. 2 · 4 2 + 1 = 2¿16 + 1 = 32 + 1 = 33 12. 1-3 · (-2) 2 ¿2 = 1 - 3¿4¿2 = 1 - 24 = -23 13 . 3^2+2^2+1 = 3 2 + 2 2 + 1 = 9 + 4 + 1 = 14 14. 2^(2^2-2) = 2 ( ) 2 2 -2 = 2 4-2 = 2 2 = 4 15. 3 - 2(-3) 2 -6(4 - 1) 2 = 3 - 2¿9 -6(3) 2 = 3 - 18 6 × 9 = -15 54 = 5 18 16. 1 - 2(1 - 4) 2 2(5 - 1) 2 · 2 = 1 - 2(-3) 2 2(4) 2 · 2 = 1 - 2¿9 2¿16¿2 = 1-18 64 = - 17 64 17. 10*(1+1/10)^3 = 10 ⎝ ⎛ ⎠ ⎞ 1 + 1 10 3 = 10(1.1) 3 = 10 × 1.331 = 13.31 18. 121/(1+1/10)^2 = 121 ⎝ ⎛ ⎠ ⎞ 1 + 1 10 2 = 121 1.1 2 = 121 1.21 = 100 19. 3 ⎣ ⎢ ⎡ ⎦ ⎥ ⎤ - 2 · 3 2 -(4 - 1) 2 = 3 ⎣ ⎢ ⎡ ⎦ ⎥ ⎤ -2 × 9 -3 2 = 3 ⎣ ⎡ ⎦ ⎤ -18 -9 = 3 × 2 = 6 20. - ⎣ ⎢ ⎡ ⎦ ⎥ ⎤ 8(1 - 4) 2 -9(5 - 1) 2 = - ⎣ ⎢ ⎡ ⎦ ⎥ ⎤ 8(-3) 2 -9(4) 2 = - ⎣ ⎡ ⎦ ⎤ 8 × 9 -9 × 16 = - ⎝ ⎛ ⎠ ⎞ 72 -144 = - ⎝ ⎛ ⎠ ⎞ - 1 2 = 1 2 21. 3 ⎣ ⎡ ⎦ ⎤ 1 - ⎝ ⎛ ⎠ ⎞ - 1 2 2 2 + 1 = 3 ⎣ ⎡ ⎦ ⎤ 1 - 1 4 2 + 1 = 3 ⎣ ⎡ ⎦ ⎤ 3 4 2 = 3 ⎣ ⎡ ⎦ ⎤ 9 16 + 1 = 27 16 + 1 = 43 16 22. 3 ⎣ ⎡ ⎦ ⎤ 1 9 - ⎝ ⎛ ⎠ ⎞ 2 3 2 2 + 1 = 3 ⎣ ⎡ ⎦ ⎤ 1 9 - 4 9 2 + 1 = 3 ⎣ ⎡ ⎦ ⎤ -3 9 2 + 1 = 3 ⎣ ⎡ ⎦ ⎤ -1 3 2 + 1 = 3 ⎣ ⎡ ⎦ ⎤ 1 9 + 1 = 3 9 + 1 = 4 3 23. (1/2)^2-1/2^2 1

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    Solutions Section 0.1 = ⎣ ⎡ ⎦ ⎤ 1 2 2 - 1 2 2 = 1 4 - 1 4 = 0 24. 2/(1^2)-(2/1)^2 = 2 1 2 - ⎣ ⎡ ⎦ ⎤ 2 1 2 = 2 1 - 4 1 = -2 25. 3¿(2-5) = 3*(2-5) 26. 4 + 5 9 = 4+5/9 or 4+(5/9) 27. 3 2-5 = 3/(2-5) Note 3/2-5 is wrong, since it corresponds to 3 2 - 5. 28. 4-1 3 = (4-1)/3 29. 3-1 8+6 = (3-1)/(8+6) Note 3-1/8-6 is wrong, since it corresponds to 3 - 1 8 - 6. 30. 3 + 3 2-9 = 3+3/(2-9) 31. 3 - 4+7 8 = 3-(4+7)/8 32. 4¿2 [ ] 2 3 = 4*2/(2/3) or (4*2)/(2/3) 33. 2 3+x - xy 2 = 2/(3+x)-x*y^2 34. 3+ 3+x xy = 3+(3+x)/(x*y) 35. 3.1x 3 - 4x -2 - 60 x 2 -1 = 3.1x^3-4x^( - 2)-60/(x^2-1) 36. 2.1x -3 - x -1 + x 2 -3 2 = 2.1x^(-3)-x^(-1)+(x^2-3)/2 37. ⎣ ⎡ ⎦ ⎤ 2 3 5 = (2/3)/5 Note that we use only (round) parentheses in technology formulas, and not brackets. 38. 2 ⎣ ⎡ ⎦ ⎤ 3 5 = 2/(3/5) 39. 3 4-5 ¿6 = 3^(4-5)*6 Note that the entire exponent is in parentheses. 40. 2 3+5 7-9 = 2/(3+5^(7-9)) 41. 3 ⎣ ⎡ ⎦ ⎤ 1 + 4 100 -3 = 3*(1+4/100)^( - 3) Note that we use only (round) parentheses in technology formulas, and not brackets. 42. 3 ⎣ ⎡ ⎦ ⎤ 1 + 4 100 -3 = 3((1+4)/100)^( - 3) 43. 3 2x-1 + 4 x - 1 = 3^(2*x-1)+4^x-1 Note that the entire exponent of 3 is in parentheses. 44. 2 x 2 - (2 2x ) 2 = 2^(x^2)-(2^(2*x))^2 45. 2 2x 2 -x+1 = 2^(2x^2-x+1) Note that the entire exponent is in parentheses. 46. 2 2x 2 -x + 1 = 2 ^(2x^2-x)+1 47. 4e -2x 2-3e -2x = 4*e^(-2*x)/(2-3e^(-2*x)) or 4(*e^(-2*x))/(2-3e^(-2*x)) or (4*e^(-2*x))/(2-3e^(-2*x)) 48. e 2x + e -2x e 2x - e -2x = (e^(2*x)+e^(-2*x))/e^(2*x)-e^(-2*x)) 49. 3 [ ] 1 - ( ) - 1 2 2 2 + 1 = 3(1-( - 1/2)^2)^2+1 Note that we use only (round) parentheses in technology formulas, and not brackets. 2

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    Solutions Section 0.2 Section 0.2 1. 3 3 = 27 2. (-2) 3 = - 8 3. -(2 · 3) 2 = -(2 2 · 3 2 ) = -(4 · 9) = -36 or -(2 · 3) 2 = -(6 2 ) = -36 4. (4 · 2) 2 = 4 2 · 2 2 = 16 · 4 = 64 or (4 · 2) 2 = 8 2 = 64 5. ⎝ ⎛ ⎠ ⎞ -2 3 2 = (-2) 2 3 2 = 4 9 6. ⎝ ⎛ ⎠ ⎞ 3 2 3 = 3 3 2 3 = 27 8 7. (-2) -3 = 1 (-2) 3 = 1 -8 = - 1 8 8. -2 -3 = - 1 2 3 = - 1 8 9. ⎝ ⎛ ⎠ ⎞ 1 4 -2 = 1 (1/4) 2 = 1 1/4 2 = 1 1/16 = 16 10. ⎝ ⎛ ⎠ ⎞ -2 3 -2 = 1 (-2/3) 2 = 1 4/9 = 9/4 11. 2 · 3 0 = 2 · 1 = 2 12. 3 · (-2) 0 = 3 · 1 = 3 13. 2 3 2 2 = 2 3+2 = 2 5 = 32 or 2 3 2 2 = 8 · 4 = 32 14. 3 2 3 = 3 2+1 = 3 3 = 27 or 3 2 3 = 9 · 3 = 27 15. 2 2 2 -1 2 4 2 -4 = 2 2-1+4-4 = 2 1 = 2 16. 5 2 5 -3 5 2 5 -2 = 5 2-3+2-2 = 5 -1 = 1 5 17. x 3 x 2 = x 3+2 = x 5 18. x 4 x -1 = x 4-1 = x 3 19. -x 2 x -3 y = -x 2-3 y = -x -1 y = - y x 20. -xy -1 x -1 = -x 1-1 y -1 = -x 0 y -1 = - 1 y 21. x 3 x 4 = x 3-4 = x -1 = 1 x 22. y 5 y 3 = y 5-3 = y 2 23. x 2 y 2 x -1 y = x 2-(-1) y 2-1 = x 3 y 24. x -1 y x 2 y 2 = x -1-2 y 1-2 = x -3 y -1 = 1 x 3 y 25. (xy -1 z 3 ) 2 x 2 yz 2 = x 2 (y -1 ) 2 (z 3 ) 2 x 2 yz 2 = x 2-2 y -2-1 z 6-2 = y -3 z 4 = z 4 y 3 26. x 2 yz 2 (xyz -1 ) -1 = x 2 yz 2 x -1 y -1 z = x 2-(-1) y 1-(-1) z 2-1 = x 3 y 2 z 27. ⎝ ⎛ ⎠ ⎞ xy -2 z x -1 z 3 = (xy -2 z) 3 (x -1 z) 3 = x 3 y -6 z 3 x -3 z 3 = x 3-(-3) y -6 z 3-3 = x 6 y -6 = x 6 y 6 28. ⎝ ⎛ ⎠ ⎞ x 2 y -1 z 0 xyz 2 = x 4 y -2 x 2 y 2 z 2 = x 4-2 y -2-2 z -2 = x 2 y -4 z -2 = x 2 y 4 z 2 29. ⎝ ⎛ ⎠ ⎞ x -1 y -2 z 2 xy -2 = (x -1-1 y -2-1 z 2 ) -2 = (x -2 y -3 z 2 ) -2 = x 4 y 6 z -4 = x 4 y 6 z 4 30. ⎝ ⎛ ⎠ ⎞ xy -2 x 2 y -1 z -3 = (x 1-2 y -2+1 z -1 ) -3 = (x -1 y -1 z -1 ) -3 = x 3 y 3 z 3 31. 3x -4 = 3 x 4 3

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    Solutions Section 0.2 32. 1 2 x -4 = 1 2x 4 33. 3 4 x -2/3 = 3 4x 2/3 34. 4 5 y -3/4 = 4 5y 3/4 35. 1 - 0.3 x -2 - 6 5 x -1 = 1 - 0.3x 2 - 6 5x 36. 1 3x -4 + 0.1x -2 3 = x 4 3 + 0.1 3x 2 37. 4 = 2 38. 5 ‡ 2.236 39. 1 4 = 1 4 = 1 2 40. 1 9 = 1 9 = 1 3 41. 16 9 = 16 9 = 4 3 42. 9 4 = 9 4 = 3 2 43. 4 5 = 2 5 44. 6 25 = 6 5 45. 9 + 16 = 3 + 4 = 7 46. 25 - 16 = 5 - 4 = 1 47. 9 + 16 = 25 = 5 48. 25 – 16 = 9 = 3 49. 3 8 – 27 = 3 –19 ‡ -2.668 50. 4 81 – 16 = 4 65 ‡ 2.839 51. 3 27 8 = 3 27 3 8 = 3 2 52. 3 8 · 64 = 3 8 · 3 64 = 2 · 4 = 8 53. (–2) 2 = 4 = 2 54. (–1) 2 = 1 = 1 55. 1 4 (1 + 15) = 16 4 = 16 4 = 4 2 = 2 56. 1 9 (3 + 33) = 36 9 = 36 9 = 6 3 = 2 57. a 2 b 2 = a b = ab 58. a 2 b 2 = a 2 b 2 = a b 59. ( x + 9) 2 = x + 9 ( x + 9 > 0 because x is positive) 60. ( x + 9 ) 2 = x + 9 61. 3 x 3 ( a 3 + b 3 ) = 3 x 3 3 a 3 + b 3 = x 3 a 3 + b 3 (Notice: Not x(a + b). ) 62. 4 x 4 a 4 b 4 = 4 x 4 4 a 4 4 b 4 = x ab 63. 4xy 3 x 2 y = 4y 2 x = 4 y 2 x = 2y x 4

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    Solutions Section 0.2 64. 4(x 2 + y 2 ) c 2 = 4 x 2 + y 2 c 2 = 2 x 2 + y 2 c (Notice: Not 2(x + y)/c .) 65. 3 = 3 1/2 66. 8 = 8 1/2 67. x 3 = x 3/2 68. 3 x 2 = x 2/3 69. 3 xy 2 = (xy 2 ) 1/3 70. x 2 y = (x 2 y) 1/2 71. x 2 x = x 2 x 1/2 = x 2-1/2 = x 3/2 72. x x = x x 1/2 = x 1-1/2 = x 1/2 73. 3 5x 2 = 3 5 x -2 74. 2 5x -3 = 2 5 x 3 75. 3x -1.2 2 - 1 3x 2.1 = 3 2 x -1.2 - 1 3 x -2.1 76. 2 3x -1.2 - x 2.1 3 = 2 3 x 1.2 - 1 3 x 2.1 77. 2x 3 - x 0.1 2 + 4 3x 1.1 = 2 3 x - 1 2 x 0.1 + 4 3 x -1.1 78. 4x 2 3 + x 3/2 6 - 2 3x 2 = 4 3 x 2 + 1 6 x 3/2 - 2 3 x -2 79. 3 x 4 - 5 3 x + 4 3x x = 3x 1/2 4 - 5 3x 1/2 + 4 3xx 1/2 = 3 4 x 1/2 - 5 3 x -1/2 + 4 3 x -3/2 80. 3 5 x - 5 x 8 + 7 2 3 x = 3 5x 1/2 - 5x 1/2 8 + 7 2x 1/3 = 3 5 x -1/2 - 5 8 x 1/2 + 7 2 x -1/3 81. 3 5 x 2 4 - 7 2 x 3 = 3x 2/5 4 - 7 2x 3/2 = 3 4 x 2/5 - 7 2 x -3/2 82. 1 8x x - 2 3 5 x 3 = 1 8x 3/2 - 2 3x 3/5 = 1 8 x -3/2 - 2 3 x -3/5 83. 1 (x 2 + 1) 3 - 3 4 3 (x 2 + 1) = 1 (x 2 + 1) 3 - 3 4(x 2 + 1) 1/3 = (x 2 + 1) -3 - 3 4 (x 2 + 1) -1/3 84. 2 3(x 2 + 1) -3 - 3 3 (x 2 + 1) 7 4 = 2 3 (x 2 + 1) 3 - 3 4 (x 2 + 1) 7/3 85. 2 2/3 = 3 2 2 86. 3 4/5 = 5 3 4 87. x 4/3 = 3 x 4 88. y 7/4 = 4 y 7 89. (x 1/2 y 1/3 ) 1/5 = 5 x 3 y 90. x -1/3 y 3/2 = y 3/2 x 1/3 = y 3 3 x 5

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    Solutions Section 0.2 91. - 3 2 x -1/4 = - 3 2x 1/4 = - 3 2 4 x 92. 4 5 x 3/2 = 4 x 3 5 93. 0.2x -2/3 + 3 7x -1/2 = 0.2 x 2/3 + 3x 1/2 7 = 0.2 3 x 2 + 3 x 7 94. 3.1 x -4/3 - 11 7 x -1/7 = 3.1x 4/3 - 11 7x 1/7 = 3.1 3 x 4 - 11 7 7 x 95. 3 4(1 - x) 5/2 = 3 4 (1 – x ) 5 96. 9 4(1 - x) -7/3 = 9(1 - x) 7/3 4 = 9 3 (1 – x ) 7 4 97. 4 -1/2 4 7/2 = 4 -1/2+7/2 = 4 3 = 64 98. 2 1/a 2 2/a = 2 1/a-2/a = 2 -1/a = 1 2 1/a 99. 3 2/3 3 -1/6 = 3 2/3-1/6 = 3 1/2 = 3 100. 2 1/3 2 -1 2 2/3 2 -1/3 = 2 1/3-1+2/3-1/3 = 2 -1/3 = 1 2 1/3 101. x 3/2 x 5/2 = x 3/2-5/2 = x -1 = 1 x 102. y 5/4 y 3/4 = y 5/4-3/4 = y 1/2 = y 103. x 1/2 y 2 x -1/2 y = x 1/2+1/2 y 2-1 = xy 104. x -1/2 y x 2 y 3/2 = x -1/2-2 y 1-3/2 = x -5/2 y -1/2 105. ⎝ ⎛ ⎠ ⎞ x y 1/3 ⎝ ⎛ ⎠ ⎞ y x 2/3 = ⎝ ⎛ ⎠ ⎞ y x -1/3 ⎝ ⎛ ⎠ ⎞ y x 2/3 = ⎝ ⎛ ⎠ ⎞ y x 1/3 106. ⎝ ⎛ ⎠ ⎞ x y -1/3 ⎝ ⎛ ⎠ ⎞ y x 1/3 = ⎝ ⎛ ⎠ ⎞ y x 1/3 ⎝ ⎛ ⎠ ⎞ y x 1/3 = ⎝ ⎛ ⎠ ⎞ y x 2/3 107. x 2 - 16 = 0, x 2 = 16, x = ± 16 = ±4 108. x 2 - 1 = 0, x 2 = 1, x = ± 1 = ±1 109. x 2 - 4 9 = 0, x 2 = 4 9 , x = ± 4 9 = ± 2 3 110. x 2 - 1 10 = 0, x 2 = 1 10 , x = ± 1 10 = ± 1 10 ‡ ±0.3162 111. x 2 - (1 + 2x) 2 = 0, x 2 = (1 + 2x) 2 , x = ±(1 + 2x); if x = 1 + 2x then -x = 1, x = -1 ; if x = -(1 + 2x) then 3x = -1, x = -1/3 . So , x = -1 or - 1/3 . 112. x 2 - (2 - 3x) 2 = 0, x 2 = (2 - 3x) 2 , x = ±(2 - 3x) ; if x = 2 - 3x then 4x = 2, x = 1/2 ; if x = -(2 - 3x) then -2x = -2, x = 1 . So, x = 1 or 1/2 . 113. x 5 + 32 = 0, x 5 = -32, x = 5 –32 = -2 114. x 4 - 81 = 0, x 4 = 81, x = ± 4 81 = ±3 115. x 1/2 - 4 = 0, x 1/2 = 4, x = 4 2 = 16 116. x 1/3 - 2 = 0, x 1/3 = 2, x = 2 3 = 8 117. 1 - 1 x 2 = 0, 1 = 1 x 2 , x 2 = 1, x = ± 1 = ±1 118. 2 x 3 - 6 x 4 = 0, 2 x 3 = 6 x 4 , 2x 4 = 6x 3 , 2x = 6, x = 3 119. (x - 4) -1/3 = 2, x - 4 = 2 -3 = 1 8 , x = 4 + 1 8 = 33 8 120. (x - 4) 2/3 + 1 = 5, (x - 4) 2/3 = 4, x - 4 = ±4 3/2 = ±8, x = 4 ± 8 = -4 or 12 6

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    Solutions Section 0.3 Section 0.3 1. x(4x + 6) = 4x 2 + 6x 2. (4y - 2)y = 4y 2 - 2y 3. (2x - y)y = 2xy - y 2 4. x(3x + y) = 3x 2 + xy 5. (x + 1)(x - 3) = x 2 + x - 3x - 3 = x 2 - 2x - 3 6. (y + 3)(y + 4) = y 2 + 3y + 4y + 12 = y 2 + 7y + 12 7. (2y + 3)(y + 5) = 2y 2 + 3y + 10y + 15 = 2y 2 + 13y + 15 8. (2x - 2)(3x - 4) = 6x 2 - 6x - 8x + 8 = 6x 2 -14x + 8 9. (2x - 3) 2 = 4x 2 - 12x + 9 10. (3x + 1) 2 = 9x 2 + 6x + 1 11. ⎝ ⎛ ⎠ ⎞ x + 1 x 2 = x 2 + 2 + 1 x 2 12. ⎝ ⎛ ⎠ ⎞ y - 1 y 2 = y 2 - 2 + 1 y 2 13. (2x - 3)(2x + 3) = (2x) 2 - 3 2 = 4x 2 - 9 14. (4 + 2x)(4 - 2x) = 4 2 - (2x) 2 = 16 - 4x 2 15. ⎝ ⎛ ⎠ ⎞ y - 1 y ⎝ ⎛ ⎠ ⎞ y + 1 y = y 2 - ⎝ ⎛ ⎠ ⎞ 1 y 2 = y 2 - 1 y 2 16. (x - x 2 )(x + x 2 ) = x 2 - (x 2 ) 2 = x 2 - x 4 17. (x 2 + x - 1)(2x + 4) = (x 2 + x - 1)2x + (x 2 + x - 1)4 = 2x 3 + 2x 2 - 2x + 4x 2 + 4x - 4 = 2x 3 + 6x 2 + 2x - 4 18. (3x + 1)(2x 2 - x + 1) = 3x(2x 2 - x + 1) + 1(2x 2 - x + 1) = 6x 3 - 3x 2 + 3x + 2x 2 - x + 1 = 6x 3 - x 2 + 2x + 1 19. (x 2 - 2x + 1) 2 = (x 2 - 2x + 1)(x 2 - 2x + 1) = x 2 (x 2 - 2x + 1) - 2x(x 2 - 2x + 1) + (x 2 - 2x + 1) = x 4 - 2x 3 + x 2 - 2x 3 + 4x 2 - 2x + x 2 - 2x + 1 = x 4 - 4x 3 + 6x 2 - 4x + 1 20. (x + y - xy) 2 = (x + y - xy)(x + y - xy) = x(x + y - xy) + y(x + y - xy) - xy(x + y - xy) = x 2 + xy - x 2 y + xy + y 2 - xy 2 - x 2 y - xy 2 + x 2 y 2 = x 2 + y 2 - 2x 2 y - 2xy 2 + 2xy + x 2 y 2 21. (y 3 + 2y 2 + y)(y 2 + 2y - 1) = y 3 (y 2 + 2y - 1) + 2y 2 (y 2 + 2y - 1) + y(y 2 + 2y - 1) = y 5 + 2y 4 - y 3 + 2y 4 + 4y 3 - 2y 2 + y 3 + 2y 2 - y = y 5 + 4y 4 + 4y 3 - y 22. (x 3 - 2x 2 + 4)(3x 2 - x + 2) = x 3 (3x 2 - x + 2) - 2x 2 (3x 2 - x + 2) + 4(3x 2 - x + 2) = 3x 5 - x 4 + 2x 3 - 6x 4 + 2x 3 - 4x 2 + 12x 2 - 4x + 8 = 3x 5 - 7x 4 + 4x 3 + 8x 2 - 4x + 8 23. (x + 1)(x + 2) + (x + 1)(x + 3) = (x + 1)(x + 2 + x + 3) = (x + 1)(2x + 5) 24. (x + 1)(x + 2) 2 + (x + 1) 2 (x + 2) = (x + 1)(x + 2)(x + 2 + x + 1) = (x + 1)(x + 2)(2x + 3) 25. (x 2 + 1) 5 (x + 3) 4 + (x 2 + 1) 6 (x + 3) 3 = (x 2 + 1) 5 (x + 3) 3 (x + 3 + x 2 + 1) = (x 2 + 1) 5 (x + 3) 3 (x 2 + x + 4) 26. 10x(x 2 + 1) 4 (x 3 + 1) 5 + 15x 2 (x 2 + 1) 5 · (x 3 + 1) 4 = 5x(x 2 + 1) 4 (x 3 + 1) 4 [2(x 3 + 1) + 3x(x 2 + 1)] = 5x(x 2 + 1) 4 (x 3 + 1) 4 (5x 3 + 3x + 2) 27. (x 3 + 1) x + 1 - (x 3 + 1) 2 x + 1 = (x 3 + 1) x + 1 [1 - (x 3 + 1)] = -x 3 (x 3 + 1) x + 1 28. (x 2 + 1) x + 1 - ( x + 1) 3 = x + 1 · [x 2 + 1 - ( x + 1) 2 ] = x + 1 [x 2 + 1 - (x + 1)] = (x 2 - x) x + 1 = x(x - 1) x + 1 29. ( x + 1) 3 + ( x + 1) 5 = ( x + 1) 3 · [1 + ( x + 1) 2 ] = ( x + 1) 3 (1 + x + 1) = 7

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    Solutions Section 0.3 (x + 2) ( x + 1) 3 30. (x 2 + 1) 3 ( x + 1) 4 - 3 ( x + 1) 7 = 3 ( x + 1) 4 [x 2 + 1 - 3 ( x + 1) 3 ] = 3 ( x + 1) 4 [x 2 + 1 - (x + 1)] = (x 2 - x) 3 ( x + 1) 4 = x(x - 1) 3 ( x + 1) 4 31. (a) 2x + 3x 2 = x(2 + 3x) (b) x(2 + 3x) = 0; x = 0 or 2 + 3x = 0; x = 0 or -2/3 32. (a) y 2 - 4y = y(y - 4) (b) y(y - 4) = 0; y = 0 or y - 4 = 0; y = 0 or 4 33. (a) 6x 3 - 2x 2 = 2x 2 (3x - 1) (b) 2x 2 (3x - 1) = 0; x = 0 or 3x - 1 = 0; x = 0 or 1/3 34. (a) 3y 3 - 9y 2 = 3y 2 (y - 3) (b) 3y 2 (y - 3) = 0; y = 0 or y - 3 = 0 ; y = 0 or 3 35. (a) x 2 - 8x + 7 = (x - 1)(x - 7) (b) (x - 1)(x - 7) = 0; x - 1 = 0 or x - 7 = 0; x = 1 or 7 36. (a) y 2 + 6y + 8 = (y + 2)(y + 4) (b) (y + 2)(y + 4) = 0; y + 2 = 0 or y + 4 = 0; y = -2 or -4 37. (a) x 2 + x - 12 = (x - 3)(x + 4) (b) (x - 3)(x + 4) = 0; x - 3 = 0 or x + 4 = 0; x = 3 or -4 38. (a) y 2 + y - 6 = (y - 2)(y + 3) (b) (y - 2)(y + 3) = 0; y - 2 = 0 or y + 3 = 0; y = 2 or -3 39. (a) 2x 2 - 3x - 2 = (2x + 1)(x - 2) (b) (2x + 1)(x - 2) = 0; 2x + 1 = 0 or x - 2 = 0; x = - 1/2 or 2 40. (a) 3y 2 - 8y - 3 = (3y + 1)(y - 3) (b) (3y + 1)(y - 3) = 0; 3y + 1 = 0 or y - 3 = 0; y = -1/3 or 3 41. (a) 6x 2 + 13x + 6 = (2x + 3)(3x + 2) (b) (2x + 3)(3x + 2) = 0; 2x + 3 = 0 or 3x + 2 = 0; x = -3/2 or -2/3 42. (a) 6y 2 + 17y + 12 = (3y + 4)(2y + 3) (b) (3y + 4)(2y + 3) = 0; 3y + 4 = 0 or 2y + 3 = 0; y = -4/3 or -3/2 43. (a) 12x 2 + x - 6 = (3x - 2)(4x + 3) (b) (3x - 2)(4x + 3) = 0; 3x - 2 = 0 or 4x + 3 = 0; x = 2/3 or -3/4 44. (a) 20y 2 + 7y - 3 = (4y - 1)(5y + 3) (b) (4y - 1)(5y + 3) = 0; 4y - 1 = 0 or 5y + 3 = 0; y = 1/4 or -3/5 45. (a) x 2 + 4xy + 4y 2 = (x + 2y) 2 (b) (x + 2y) 2 = 0; x + 2y = 0; x = -2y 46. (a) 4y 2 - 4xy + x 2 = (2y - x) 2 (b) (2y - x) 2 = 0; 2y - x = 0; y = x/2 47. (a) x 4 - 5x 2 + 4 = (x 2 - 1)(x 2 - 4) = (x - 1)(x + 1)(x - 2)(x + 2) (b) (x - 1)(x + 1)(x - 2)(x + 2) = 0; x - 1 = 0 or x + 1 = 0 or x - 2 = 0 or x + 2 = 0; x = ±1 or ±2 48. (a) y 4 + 2y 2 - 3 = (y 2 - 1)(y 2 + 3) = (y - 1)(y + 1)(y 2 + 3) (b) (y - 1)(y + 1)(y 2 + 3) = 0; y - 1 = 0 or y + 1 = 0 or y 2 + 3 = 0; y = ±1 (Notice that y 2 + 3 = 0 has no real solutions.) 8

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    Solutions Section 0.4 Section 0.4 1 . x - 4 x + 1 · 2x + 1 x - 1 = (x - 4)(2x + 1) (x + 1)(x - 1) = 2x 2 - 7x - 4 x 2 - 1 2. 2x - 3 x - 2 · x + 3 x + 1 = (2x - 3)(x + 3) (x - 2)(x + 1) = 2x 2 + 3x - 9 x 2 - x - 2 3. x - 4 x + 1 + 2x + 1 x - 1 = (x - 4)(x - 1) + (x + 1)(2x + 1) (x + 1)(x - 1) = 3x 2 - 2x + 5 x 2 - 1 4. 2x - 3 x - 2 + x + 3 x + 1 = (2x - 3)(x + 1) + (x - 2)(x + 3) (x - 2)(x + 1) = 3x 2 - 9 x 2 - x - 2 5. x 2 x + 1 - x - 1 x + 1 = x 2 - (x - 1) x + 1 = x 2 - x + 1 x + 1 6. x 2 - 1 x - 2 - 1 x - 1 = (x 2 - 1)(x - 1) - (x - 2) (x - 2)(x - 1) = x 3 - x 2 - 2x + 3 x 2 - 3x + 2 7. 1 ⎝ ⎛ ⎠ ⎞ x x - 1 + x - 1 = x - 1 x + x - 1 = x - 1 + x(x - 1) x = x 2 - 1 x 8. 2 ⎝ ⎛ ⎠ ⎞ x - 2 x 2 - 1 x - 2 = 2x 2 x - 2 - 1 x - 2 = 2x 2 - 1 x - 2 9. 1 x ⎝ ⎛ ⎠ ⎞ x - 3 xy + 1 y = 1 x ⎝ ⎛ ⎠ ⎞ x - 3 + x xy = 2x - 3 x 2 y 10. y 2 x ⎝ ⎛ ⎠ ⎞ 2x - 3 y + x y = y 2 x ⎝ ⎛ ⎠ ⎞ 2x - 3 + x y = y 2 (3x - 3) xy = y(3x - 3) x = 3xy - 3y x 11. (x + 1) 2 (x + 2) 3 - (x + 1) 3 (x + 2) 2 (x + 2) 6 = (x + 1) 2 (x + 2) 2 [(x + 2) - (x + 1)] (x + 2) 6 = (x + 1) 2 (x + 2) 4 12. 6x(x 2 + 1) 2 (x 3 + 2) 3 - 9x 2 (x 2 + 1) 3 (x 3 + 2) 2 (x 3 + 2) 6 = 3x(x 2 + 1) 2 (x 3 + 2) 2 [2(x 3 + 2) - 3x(x 2 + 1)] (x 3 + 2) 6 = 3x(x 2 + 1) 2 (-x 3 - 3x + 4) (x 3 + 2) 4 = -3x(x 2 + 1) 2 (x 3 + 3x - 4) (x 3 + 2) 4 13. (x 2 - 1) x 2 + 1 - x 4 x 2 + 1 x 2 + 1 = (x 2 - 1)(x 2 + 1) - x 4 (x 2 + 1) x 2 + 1 = -1 ( x 2 + 1) 3 14. x x 3 – 1 - 3x 4 x 3 – 1 x 3 - 1 = x(x 3 - 1) - 3x 4 (x 3 - 1) x 3 – 1 = -2x 4 - x ( x 3 – 1) 3 = -x(2x 3 + 1) ( x 3 – 1) 3 15. 1 (x + y) 2 - 1 x 2 y = x 2 - (x + y) 2 yx 2 (x + y) 2 = x 2 - x 2 - 2xy - y 2 yx 2 (x + y) 2 = -y(2x + y) yx 2 (x + y) 2 = -(2x + y) x 2 (x + y) 2 16. 1 (x + y) 3 - 1 x 3 y = x 3 - (x + y) 3 yx 3 (x + y) 3 = x 3 - x 3 - 3x 2 y - 3xy 2 - y 3 yx 3 (x + y) 3 = -y(3x 2 + 3xy + y 2 ) yx 3 (x + y) 3 = -(3x 2 + 3xy + y 2 ) x 3 (x + y) 3 9

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    Solutions Section 0.5 Section 0.5 1. x + 1 = 0, x = -1 2. x - 3 = 1, x = 4 3. -x + 5 = 0, x = 5 4. 2x + 4 = 1, 2x = -3, x = -3/2 5. 4x - 5 = 8, 4x = 13, x = 13/4 6. 3 4 x + 1 = 0, 3 4 x = -1, x = -4/3 7. 7x + 55 = 98, 7x = 43, x = 43/7 8. 3x + 1 = x, 2x = -1, x = -1/2 9. x + 1 = 2x + 2, -x = 1, x = -1 10. x + 1 = 3x + 1, -2x = 0, x = 0 11. ax + b = c, ax = c - b, x = (c - b)/a 12. x - 1 = cx + d, (1 - c)x = d + 1, x = d + 1 1 - c 13. 2x 2 + 7x - 4 = 0, (2x - 1)(x + 4) = 0, x = -4, 1 2 14. x 2 + x + 1 = 0, ∆ = -3 < 0 , so this equation has no real solutions 15. x 2 - x + 1 = 0, ∆ = -3 < 0 , so this equation has no real solutions 16. 2x 2 - 4x + 3 = 0, ∆ = -8 < 0 , so this equation has no real solutions 17. 2x 2 - 5 = 0, x 2 = 5 2 , x = ± 5 2 18. 3x 2 - 1 = 0, x 2 = 1 3 , x = ± 1 3 19. -x 2 - 2x - 1 = 0, -(x + 1) 2 = 0, x = -1 20. 2x 2 - x - 3 = 0, (2x - 3)(x + 1) = 0, x = 3/2, - 1 21. 1 2 x 2 - x - 3 2 = 0, x 2 - 2x - 3 = 0, (x + 1)(x - 3) = 0, x = -1, 3 22. - 1 2 x 2 - 1 2 x + 1 = 0, x 2 + x - 2 = 0, (x + 2)(x - 1) = 0, x = -2, 1 23. x 2 - x = 1, x 2 - x - 1 = 0, x = 1 ± 5 2 by the quadratic formula 24. 16x 2 = -24x - 9, 16x 2 + 24x + 9 = 0, (4x + 3) 2 = 0, x = - 3/4 25. x = 2 - 1 x , x 2 = 2x - 1, x 2 - 2x + 1 = 0, (x - 1) 2 = 0, x = 1 26. x + 4 = 1 x - 2 , (x + 4)(x - 2) = 1, x 2 + 2x - 8 = 1, x 2 + 2x - 9 = 0, x = -2 ± 40 2 = -1 ± 10 by the quadratic formula 27. x 4 - 10x 2 + 9 = 0, (x 2 - 1)(x 2 - 9) = 0, x 2 = 1 or x 2 = 0, x = ±1, ±3 28. x 4 - 2x 2 + 1 = 0, (x 2 - 1) 2 = 0, x = ±1 29. x 4 + x 2 - 1 = 0, x 2 = -1 ± 5 2 by the quadratic formula, x = ± -1 ± 5 2 30. x 3 + 2x 2 + x = 0, x(x 2 + 2x + 1) = 0, x(x + 1) 2 = 0, x = 0, -1 31. x 3 + 6x 2 + 11x + 6 = 0, (x + 1)(x + 2)(x + 3) = 0, x = -1, -2, -3 32. x 3 - 6x 2 + 12x - 8 = 0, (x - 2) 3 = 0, x = 2 33. x 3 + 4x 2 + 4x + 3 = 0, 10

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    Solutions Section 0.5 (x + 3)(x 2 + x + 1) = 0, x = -3 (For x 2 + x + 1 = 0, ∆ = -3 < 0 , so there are no real solutions to this quadratic equation.) 34. y 3 + 64 = 0, y 3 = -64, y = 3 –64 = -4 35. x 3 - 1 = 0, x 3 = 1, x = 3 1 = 1 36. x 3 - 27 = 0, x 3 = 27, x = 3 27 = 3 37. y 3 + 3y 2 + 3y + 2 = 0, (y + 2)(y 2 + y + 1) = 0, y = -2 (For y 2 + y + 1 = 0, ∆ = -3 < 0 , so there are no real solutions to this quadratic equation.) 38. y 3 - 2y 2 - 2y - 3 = 0, (y - 3)(y 2 + y + 1) = 0, y = 3 (For y 2 + y + 1 = 0, ∆ = -3 < 0 , so there are no real solutions to this quadratic equation.) 39. x 3 - x 2 - 5x + 5 = 0, (x - 1)(x 2 - 5) = 0, x = 1, ± 5 40. x 3 - x 2 - 3x + 3 = 0, (x - 1)(x 2 - 3) = 0, x = 1, ± 3 41. 2x 6 - x 4 - 2x 2 + 1 = 0, (2x 2 - 1)(x 4 - 1) = 0, [ or (2x 2 - 1)(x 2 - 1)(x 2 + 1) = 0 ; in any case, think of the cubic you get by substituting y for x 2 ], x = ±1, ± 1 2 42. 3x 6 - x 4 - 12x 2 + 4 = 0, (3x 2 - 1)(x 4 - 4) = 0, [ or (3x 2 - 1)(x 2 - 2)(x 2 + 2) = 0], x = ± 2 , ± 1 3 43. (x 2 + 3x + 2)(x 2 - 5x + 6) = 0, (x + 2)(x + 1)(x - 2)(x - 3) = 0, x = -2, -1, 2, 3 44. (x 2 - 4x + 4) 2 (x 2 + 6x + 5) 3 = 0, (x - 2) 4 (x + 1) 3 (x + 5) 3 = 0, x = -5, -1, 2 11

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    Solutions Section 0.6 Section 0.6 1. x 4 - 3x 3 = 0, x 3 (x - 3) = 0, x = 0, 3 2. x 6 - 9x 4 = 0, x 4 (x 2 - 9) = 0, x = 0, ±3 3. x 4 - 4x 2 = -4, x 4 - 4x 2 + 4 = 0, (x 2 - 2) 2 = 0, x = ± 2 4. x 4 - x 2 = 6, x 4 - x 2 - 6 = 0, (x 2 - 3)(x 2 + 2) = 0, x = ± 3 5. (x + 1)(x + 2) + (x + 1)(x + 3) = 0, (x + 1)(x + 2 + x + 3) = 0, (x + 1)(2x + 5) = 0, x = -1, -5/2 6. (x + 1)(x + 2) 2 + (x + 1) 2 (x + 2) = 0, (x + 1)(x + 2)(x + 2 + x + 1) = 0, (x + 1)(x + 2)(2x + 3) = 0, x = -1, -2, -3/2 7. (x 2 + 1) 5 (x + 3) 4 + (x 2 + 1) 6 (x + 3) 3 = 0, (x 2 + 1) 5 (x + 3) 3 (x + 3 + x 2 + 1) = 0, (x 2 + 1) 5 (x + 3) 3 (x 2 + x + 4) = 0, x = -3 (Neither x 2 + 1 = 0 nor x 2 + x + 4 = 0 has a real solution.) 8. 10x(x 2 + 1) 4 (x 3 + 1) 5 - 10x 2 (x 2 + 1) 5 (x 3 + 1) 4 = 0, 10x(x 2 + 1) 4 (x 3 + 1) 4 [x 3 + 1 - x(x 2 + 1)] = 0, 10x(x 2 + 1) 4 (x 3 + 1) 4 (1 - x) = 0, x = -1, 0, 1 9. (x 3 + 1) x + 1 - (x 3 + 1) 2 x + 1 = 0, (x 3 + 1) x + 1 [1 - (x 3 + 1)] = 0, -x 3 (x 3 + 1) x + 1 = 0, x = 0, -1 10. (x 2 + 1) x + 1 - ( x + 1) 3 = 0, x + 1 [x 2 + 1 - (x + 1)] = 0, (x 2 - x) x + 1 = 0, x(x - 1) x + 1 = 0, x = -1, 0, 1 11. ( x + 1) 3 + ( x + 1) 5 = 0, ( x + 1) 3 (1 + x + 1) = 0, (x + 2) ( x + 1) 3 = 0, x = -1 (x = -2 is not a solution because ( x + 1) 3 is not defined for x = -2 .) 12. (x 2 + 1) 3 ( x + 1) 4 - 3 ( x + 1) 7 = 0, 3 ( x + 1) 4 [x 2 + 1 - (x + 1)] = 0, (x 2 - x) 3 ( x + 1) 4 = 0, x(x - 1) 3 ( x + 1) 4 = 0, x = -1, 0, 1 13. (x + 1) 2 (2x + 3) - (x + 1)(2x + 3) 2 = 0, (x + 1)(2x + 3)(x + 1 - 2x - 3) = 0, (x + 1)(2x + 3)(-x - 2) = 0, x = -2, -3/2, -1 14. (x 2 - 1) 2 (x + 2) 3 - (x 2 - 1) 3 (x + 2) 2 = 0, (x 2 - 1) 2 (x + 2) 2 (x + 2 - x 2 + 1) = 0, -(x 2 - 1) 2 (x + 2) 2 (x 2 - x - 3) = 0, x = -2, -1, 1, (1 ± 13 )/2 15. (x + 1) 2 (x + 2) 3 - (x + 1) 3 (x + 2) 2 (x + 2) 6 = 0, (x + 1) 2 (x + 2) 2 [(x + 2) - (x + 1)] (x + 2) 6 = 0, (x + 1) 2 (x + 2) 4 = 0, (x + 1) 2 = 0, x = -1 16. 6x(x 2 + 1) 2 (x 2 + 2) 4 - 8x(x 2 + 1) 3 (x 2 + 2) 3 (x 2 + 2) 8 = 0, 2x(x 2 + 1) 2 (x 2 + 2) 3 [3(x 2 + 2) - 4(x 2 + 1)] (x 2 + 2) 8 = 0, -2x(x 2 + 1) 2 (x 2 - 2) (x 2 + 2) 5 = 0, -2x(x 2 + 1) 2 (x 2 - 2) = 0, x = 0, ± 2 17. 2(x 2 - 1) x 2 + 1 - x 4 x 2 + 1 x 2 + 1 = 0, 2(x 2 - 1)(x 2 + 1) - x 4 (x 2 + 1) x 2 + 1 = 0, x 4 - 2 (x 2 + 1) x 2 + 1 = 0, x 4 - 2 = 0, x = ± 4 2 18. 4x x 3 – 1 - 3x 4 x 3 – 1 x 3 - 1 = 0, 4x(x 3 - 1) - 3x 4 (x 3 - 1) x 3 – 1 = 0, x 4 - 4x (x 3 - 1) x 3 - 1 = 0, x 4 - 4x = 0, x(x 3 - 4) = 0, x = 0, ± 3 4 19. x - 1 x = 0, x 2 - 1 = 0, x = ±1 12

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    Solutions Section 0.6 20. 1 - 4 x 2 = 0, x 2 - 4 = 0, x = ±2 21. 1 x - 9 x 3 = 0, x 2 - 9 = 0, x = ±3 22. 1 x 2 - 1 x + 1 = 0, x + 1 - x 2 = 0, x 2 - x - 1 = 0, x = (1 ± 5) /2 23. x - 4 x + 1 - x x - 1 = 0, (x - 4)(x - 1) - x(x + 1) (x + 1)(x - 1) = 0, -6x + 4 (x + 1)(x - 1) = 0, -6x + 4 = 0, x = 2/3 24. 2x - 3 x - 1 - 2x + 3 x + 1 = 0, (2x - 3)(x + 1) - (2x + 3)(x - 1) (x - 1)(x + 1) = 0, -2x (x - 1)(x + 1) = 0, -2x = 0, x = 0 25. x + 4 x + 1 + x + 4 3x = 0, 3x(x + 4) + (x + 1)(x + 4) 3x(x + 1) = 0, (x + 4)(3x + x + 1) 3x(x + 1) = 0, (x + 4)(4x + 1) 3x(x + 1) = 0, (x + 4)(4x + 1) = 0, x = -4, -1/4 26. 2x - 3 x - 2x - 3 x + 1 = 0, (2x - 3)(x + 1) - x(2x - 3) x(x + 1) = 0, (2x - 3)(x + 1 - x) x(x + 1) = 0, 2x - 3 x(x + 1) = 0, 2x - 3 = 0, x = 3/2 13

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