SECTION 4.3 The Real Zeros of a Polynomial Function 227 Skill Building In Problems 11–20, use the Remainder Theorem to find the remainder when f x( ) is divided by x c − . Then use the Factor Theorem to determine whether x c − is a factor of f x( ). 11. f x x x x x 4 3 8 4; 2 3 2 ( ) = − − + − 12. f x x x x 4 5 8; 3 3 2 ( ) = − + + + 13. f x x x x x 5 20 4; 2 4 3 ( ) = − + − − 14. f x x x x 4 15 4; 2 4 2 ( ) = − − − 15. f x x x x 2 129 64; 4 6 3 ( ) = + + + 16. ( ) = − + − + f x x x x x 2 18 9; 3 6 4 2 17. f x x x x x 4 64 15; 4 6 4 2 ( ) = − + − + 18. ( ) = − + − + f x x x x x 16 16; 4 6 4 2 19. f x x x x x 2 2 1; 1 2 4 3 ( ) = − + − − 20. f x x x x x 3 3 1; 1 3 4 3 ( ) = + − + + In Problems 21–32, determine the maximum number of real zeros that each polynomial function may have.Then use Descartes’ Rule of Signs to determine how many positive and how many negative real zeros each polynomial function may have. Do not attempt to find the zeros. 21. f x x x x 4 2 7 3 2 ( ) = − + − + 22. f x x x x 5 2 6 5 4 2 ( ) = + − − 23. f x x x x 8 7 5 6 2 ( ) = − − + 24. f x x x 3 4 2 5 4 ( ) = − + + 25. f x x x x 2 5 7 3 2 ( ) = − + − − 26. f x x x x 1 3 2 ( ) = − − + + 27. f x x x 1 4 2 ( ) = − + − 28. f x x x5 2 4 3 ( ) = + − 29. f x x x x x 1 5 4 2 ( ) = + + + + 30. ( ) = − + − + − f x x x x x x 1 5 4 3 2 31. f x x 1 6 ( ) = − 32. f x x 1 6 ( ) = + In Problems 33–44, list the potential rational zeros of each polynomial function. Do not attempt to find the zeros. 33. f x x x x x 3 3 1 4 3 2 ( ) = − + − + 34. f x x x x2 3 5 4 2 ( ) = − + + 35. f x x x x 2 8 5 5 2 ( ) = − + − 36. f x x x x 2 1 5 4 2 ( ) = − − + 37. f x x x x 9 3 3 2 ( ) = − − + + 38. f x x x 6 2 4 2 ( ) = − + 39. f x x x 6 9 4 2 ( ) = − + 40. f x x x x 4 6 3 2 ( ) = − + + + 41. f x x x x 2 2 12 5 3 2 ( ) = − + + 42. f x x x x 3 2 18 5 2 ( ) = − + + 43. f x x x x 6 2 20 4 3 2 ( ) = + − + 44. f x x x x 6 10 3 2 ( ) = − − + + In Problems 63–68, find the real zeros of f. If necessary, round to two decimal places. 63. f x x x x 3.2 16.83 5.31 3 2 ( ) = + − − 64. f x x x x 3.2 7.25 6.3 3 2 ( ) = + − − 65. ( ) = − − + + f x x x x x 1.4 33.71 23.94 292.41 4 3 2 66. ( ) = + − − + f x x x x x 1.2 7.46 4.692 15.2881 4 3 2 67. f x x x x 19.5 1021 1000.5 3 2 ( ) = + − + 68. ( ) = + − + f x x x x 42.2 664.8 1490.4 3 2 In Problems 45–62, find the real zeros of f. Use the real zeros to factor f. 45. f x x x x 2 5 6 3 2 ( ) = + − − 46. f x x x x 8 11 20 3 2 ( ) = + + − 47. f x x x x 2 13 24 9 3 2 ( ) = − + − 48. f x x x x 2 5 4 12 3 2 ( ) = − − + 49. f x x x x 3 4 4 1 3 2 ( ) = + + + 50. f x x x x 3 7 12 28 3 2 ( ) = − + − 51. f x x x x 10 28 16 3 2 ( ) = − + − 52. f x x x x 6 6 4 3 2 ( ) = + + − 53. f x x x x x 3 2 4 3 2 ( ) = + − − + 54. f x x x x x 6 4 8 4 3 2 ( ) = − − + + 55. f x x x x x 21 22 99 72 28 4 3 2 ( ) = + − − + 56. f x x x x x 54 57 323 20 12 4 3 2 ( ) = − − + + 57. f x x x x 8 17 6 3 2 ( ) = − + − 58. f x x x x x 2 11 5 43 35 4 3 2 ( ) = + − − + 59. f x x x 4 7 2 4 2 ( ) = + − 60. f x x x 4 15 4 4 2 ( ) = + − 61. f x x x x 4 8 2 5 4 ( ) = − − + 62. f x x x x 4 12 3 5 4 ( ) = + − − In Problems 69–80, solve each equation in the real number system. 69. x x x x 2 4 8 0 4 3 2 − + − − = 70. x x x 2 3 2 3 0 3 2 + + + = 71. x x x 3 4 7 2 0 3 2 + − + = 72. x x x 2 3 3 5 0 3 2 − − − = 73. x x x 3 15 5 0 3 2 − − + = 74. x x x 2 11 10 8 0 3 2 − + + = 75. x x x x 4 2 6 0 4 3 2 + + − + = 76. x x x x 2 10 18 9 0 4 3 2 − + − + = 77. x x x 2 3 8 3 1 0 3 2 − + + = 78. x x x 3 2 3 2 0 3 2 + + − = 79. x x x x 2 19 57 64 20 0 4 3 2 − + − + = 80. x x x x 2 24 20 16 0 4 3 2 + − + + =
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