346 CHAPTER 3 Polynomial and Rational Functions 19. x4 - 3x3 - 4x2 + 12x x - 2 20. x4 - x3 - 5x2 - 3x x + 1 21. x3 - 1 x - 1 22. x4 - 1 x - 1 23. x5 + 1 x + 1 24. x7 + 1 x + 1 Use synthetic division to divide ƒ1x2 by x - k for the given value of k. Then express ƒ1x2 in the form ƒ1x2 = 1x - k2q1x2 + r. 25. ƒ1x2 = 2x3 + x2 + x - 8; k = -1 26. ƒ1x2 = 2x3 + 3x2 - 16x + 10; k = -4 27. ƒ1x2 = x3 + 4x2 + 5x + 2; k = -2 28. ƒ1x2 = -x3 + x2 + 3x - 2; k = 2 29. ƒ1x2 = 4x4 - 3x3 - 20x2 - x; k = 3 30. ƒ1x2 = 2x4 + x3 - 15x2 + 3x; k = -3 31. ƒ1x2 = 3x4 + 4x3 - 10x2 + 15; k = -1 32. ƒ1x2 = -5x4 + x3 + 2x2 + 3x + 1; k = 1 For each polynomial function, use the remainder theorem to find ƒ1k2. See Example 2. 33. ƒ1x2 = x2 + 5x + 6; k = -2 34. ƒ1x2 = x2 - 4x - 5; k = 5 35. ƒ1x2 = 2x2 - 3x - 3; k = 2 36. ƒ1x2 = -x3 + 8x2 + 63; k = 4 37. ƒ1x2 = x3 - 4x2 + 2x + 1; k = -1 38. ƒ1x2 = 2x3 - 3x2 - 5x + 4; k = 2 39. ƒ1x2 = x2 - 5x + 1; k = 2 + i 40. ƒ1x2 = x2 - x + 3; k = 3 - 2i 41. ƒ1x2 = x2 + 4; k = 2i 42. ƒ1x2 = 2x2 + 10; k = i25 43. ƒ1x2 = 2x5 - 10x3 - 19x2 - 50; k = 3 44. ƒ1x2 = x4 + 6x3 + 9x2 + 3x - 3; k = 4 45. ƒ1x2 = 6x4 + x3 - 8x2 + 5x + 6; k = 1 2 46. ƒ1x2 = 6x3 - 31x2 - 15x; k = -1 2 Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ1k2. See Examples 2 and 3. 47. ƒ1x2 = x2 + 2x - 8; k = 2 48. ƒ1x2 = x2 + 4x - 5; k = -5 49. ƒ1x2 = x3 - 3x2 + 4x - 4; k = 2 50. ƒ1x2 = x3 + 2x2 - x + 6; k = -3 51. ƒ1x2 = 2x3 - 6x2 - 9x + 4; k = 1 52. ƒ1x2 = 2x3 + 9x2 - 16x + 12; k =1 53. ƒ1x2 = x3 + 7x2 + 10x; k = 0 54. ƒ1x2 = 2x3 - 3x2 - 5x; k = 0 55. ƒ1x2 = 5x4 + 2x3 - x + 3; k = 2 5 56. ƒ1x2 = 16x 4 + 3x2 - 2; k = 1 2 57. ƒ1x2 = x2 - 2x + 2; k = 1 - i 58. ƒ1x2 = x2 - 4x + 5; k = 2 - i 59. ƒ1x2 = x2 + 3x + 4; k = 2 + i 60. ƒ1x2 = x2 - 3x + 5; k = 1 - 2i 61. ƒ1x2 = 4x4 + x2 + 17x + 3; k = -3 2 62. ƒ1x2 = 3x4 + 13x3 - 10x + 8; k = -4 3 63. ƒ1x2 = x3 + 3x2 - x + 1; k = 1 + i 64. ƒ1x2 = 2x3 - x2 + 3x - 5; k = 2 - i
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