A-16 Answers to Selected Exercises 3.3 Exercises 1. true 3. false; -2 is a zero of multiplicity 4. The number 2 is not a zero. 5. true 7. false; z = 7 + 6i 9. yes 11. no 13. yes 15. no 17. no 19. yes 21. ƒ1x2 = 1x - 2212x - 521x + 32 23. ƒ1x2 = 1x + 3213x - 1212x - 12 25. ƒ1x2 = -1x + 4213x - 1212x + 12 27. ƒ1x2 = 1x - 3i21x + 421x + 32 29. ƒ1x2 = 3x - 11 + i2412x - 121x + 32 31. ƒ1x2 = 1x + 222 1x + 121x - 32 33. -1{i 35. 3, 2 + i 37. i, {2i 39. (a) {1, {2, {5, {10 (b) -1, -2, 5 (c) ƒ1x2 = 1x + 121x + 221x - 52 41. (a) {1, {2, {3, {5, {6, {10, {15, {30 (b) -5, -3, 2 (c) ƒ1x2 = 1x + 521x + 321x - 22 43. (a) {1, {2, {3, {4, {6, {12, { 1 2 , { 3 2 , { 1 3 , { 2 3 , { 4 3 , { 1 6 (b) -4, - 1 3 , 3 2 (c) ƒ1x2 = 1x + 4213x + 1212x - 32 45. (a) {1, {2, {3, {4, {6, {12, { 1 2 , { 3 2 , { 1 3 , { 2 3 , { 4 3 , { 1 4 , { 3 4 , { 1 6 , { 1 8 , { 3 8 , { 1 12 , { 1 24 (b) - 3 2 , - 2 3 , 1 2 (c) ƒ1x2 = 212x + 3213x + 2212x - 12 47. 2 (multiplicity 3), {17 49. 0, 2, -3, 1, -1 51. -2 (multiplicity 5), 1 (multiplicity 5), 1 - 13 (multiplicity 2) 53. ƒ1x2 = -3x3 + 6x2 + 33x - 36 55. ƒ1x2 = - 1 2 x 3 - 1 2 x 2 + x 57. ƒ1x2 = 1 6 x 3 + 3 2 x 2 + 9 2 x + 9 2 59. ƒ1x2 = 5x3 - 10x2 + 5x In Exercises 61–77, we give only one possible answer. 61. ƒ1x2 = x2 - 10x + 26 63. ƒ1x2 = x5 - 2x4 + 3x3 - 2x2 + 2x 65. ƒ1x2 = x3 - 3x2 + x + 1 67. ƒ1x2 = x4 - 6x3 + 10x2 + 2x - 15 69. ƒ1x2 = x3 - 8x2 + 22x - 20 71. ƒ1x2 = x4 - 4x3 + 5x2 - 2x - 2 73. ƒ1x2 = x4 - 16x3 + 98x2 - 240x + 225 75. ƒ1x2 = x5 - 12x4 + 74x3 - 248x2 + 445x - 500 77. ƒ1x2 = x4 - 6x3 + 17x2 - 28x + 20 79. Positive Negative Nonreal Complex 2 1 0 0 1 2 81. Positive Negative Nonreal Complex 3 0 0 1 0 2 83. Positive Negative Nonreal Complex 1 1 2 85. Positive Negative Nonreal Complex 4 0 0 2 0 2 0 0 4 87. Positive Negative Nonreal Complex 2 3 0 2 1 2 0 3 2 0 1 4 89. Positive Negative Nonreal Complex 2 3 0 2 1 2 0 3 2 0 1 4 91. Positive Negative Nonreal Complex 4 2 0 4 0 2 2 2 2 2 0 4 0 2 4 0 0 6 93. Positive Negative Nonreal Complex 0 5 0 0 3 2 0 1 4 74. x y 12 2 –8 2 –2 (–1, –2) (2, –8) (–3, 12) f(x) = –x3 – x2 + 2x ( ) , 1 2 5 8 63. no; -2 + 7i 65. -12; 1-2, -122 66. 0; 1-1, 02 67. 15 8 ; A - 1 2 , 15 8 B 68. 2; 10, 22 69. 0; 11, 02 70. - 5 8 ; A 3 2 , - 5 8B 71. 0; 12, 02 72. 8; 13, 82 73. f(x) = x3 – 2x2 – x + 2 (3, 8) (–2, –12) y x 5 8 ( , – ) 3 2 15 8 (– , ) 1 2 –2 12 0 2 95. -5, 3, {2i23 97. 1, 1, 1, -4 99. -3, -3, 0, 1 {i231 4 101. 2, 2, 2, {i 22 103. - 1 2 , 1, {2i 105. - 1 5 , 1{i 25 107. {2i, {5i 109. {i, {i 111. 0, 0, 3 {22 113. 3, 3, 1{i27 115. {2, {3, {2i 121. 2
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