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06_Chapter 4 - Polynomial and Rational Functions

06_Chapter 4 - Polynomial and Rational Functions - SECTION...

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SECTION 4-1 211 CHAPTER 4 Polynomial and Rational Functions Section 4-1 1. The degree is odd and the coefficient is positive, so f ( x ) increases without bound as x ± ± and decreases without bound as x ± - ± . This matches graph c. 3. The degree is even and the coefficient is positive, so h ( x ) increases without bound both as x ± ± and x ± - ± . This matches graph d. 5. The real zeros are the x -intercepts: -1 and 3. The turning point is (1, 4). P ( x ) ± - ± as x ± ± and P ( x ) ± - ± as x ± - ± . 7. The real zeros are the x -intercepts: -2 and 1. The turning points are (-1, 4) and (1, 0). P ( x ) ± - ± as x ± - ± and P ( x ) ± ± as x ± ± . 9. The graph of a polynomial always increases or decreases without bound as x ± ± and x ± - ± . This graph does not. 11. The graph of a polynomial always has a finite number of turning points--at most one less than the degree. This graph has infinitely many turning points. 13. Set each factor equal to zero: x = 0 x 2 - 9 = 0 x 2 + 4 = 0 x 2 = 9 x 2 = -4 x = 3, -3 x = 2 i , -2 i The zeros are 0, 3, -3, 2 i , and -2 i ; of these 0, 3, and -3 are x -intercepts. 15. Set each factor equal to zero: x + 5 = 0 x 2 + 9 = 0 x 2 + 16 = 0 x = -5 x 2 = -9 x 2 = -16 x = 3 i , -3 i x = 4 i , -4 i The zeros are -5, 3 i , -3 i , 4 i , and 4 i . Only -5 is an x intercept. 17. 3 x + 2 x + 1 3 x 2 + 5 x + 6 3 x 2 + 3 x 2 x + 6 2 x + 2 4 3 x + 2, R = 4 19. 2 m + 1 2 m - 1 4 m 2 + 0 m - 1 4 m 2 - 2 m 2 m - 1 2 m - 1 0 2 m + 1, R = 0 21. 4 x - 5 2 x + 1 8 x 2 - 6 x + 6 8 x 2 + 4 x - 10 x + 6 - 10 x - 5 11 4 x - 5, R = 11 23. x 2 + x + 1 x - 1 x 3 - 0 x 2 + 0 x - 1 x 3 - x 2 x 2 + 0 x x 2 - x x - 1 x - 1 0 x 2 + x + 1, R = 0
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212 CHAPTER 4 POLYNOMIAL AND RATIONAL FUNCTIONS 25. 2 y 2 - 5 y + 13 y + 2 2 y 3 - y 2 + 3 y - 1 2 y 3 + 4 y 2 - 5 y 2 + 3 y - 5 y 2 - 10 y 13 y - 1 13 y + 26 - 27 2 y 2 - 5 y + 13, R = -27 27. 1 3 -7 29. 4 10 -9 2 10 -12 6 2 1 5 3 -3 4 -2 -3 x 2 + 3 x ± 7 x ± 2 = x + 5 + 3 x ± 2 4 x 2 + 10 x ± 9 x + 3 = 4 x - 2 - 3 x + 3 31. 2 0 -3 1 4 8 10 2 2 4 5 11 2 x 3 ± 3 x + 1 x ± 2 = 2 x 2 + 4 x + 5 + 11 x ± 2 Common Error: The first row is not 2 -3 1 The 0 must be inserted for the missing power. 33. 1 4 -221 35. 2 38 -1 19 -17 221 -38 0 19 -17 1 -13 0 -19 2 0 -1 38 The remainder is 0, hence -17 is a zero. The remainder is non-zero, hence -19 is not a zero. 37. x - 1 will be a factor of P ( x ) if P (1) = 0. Since P ( x ) = x 18 - 1, P (1) = 1 18 - 1 = 0. Therefore x - 1 is a factor of x 18 - 1. 39. x + 1 will be a factor of P ( x ) if P (-1) = 0. Since P ( x ) = 3 x 3 - 7 x 2 - 8 x + 2, P (-1) = 3(-1) 3 - 7(-1) 2 - 8(-1) + 2 = -3 - 7 + 8 + 2 = 0. Therefore x + 1 is a factor of 3 x 3 - 7 x 2 - 8 x + 2. 41. 3 -1 -10 43. 2 -5 7 -7 -6 14 4 -2 10 -2 3 -7 4 2 2 -1 5 3 P (-2) = 4 P (2) = 3 45. 1 0 -10 25 -2 -4 16 -24 -4 -4 1 -4 6 1 -6 P (-4) = -6 47. 3 0 0 -1 -4 49. 1 0 0 0 0 1 -3 3 -3 4 -1 1 -1 1 -1 -1 3 -3 3 -4 0 -1 1 -1 1 -1 1 0 3 x 3 - 3 x 2 + 3 x - 4, R = 0 x 4 - x 3 + x 2 - x + 1, R = 0
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SECTION 4-1 213 51. 3 2 0 -4 -1 53. 2 -13 0 75 2 0 -50 -9 21 -63 201 10 -15 -75 0 10 50 -3 3 -7 21 -67 200 5 2 -3 -15 0 2 10 0 3 x 3 - 7 x 2 + 21 x - 67, R = 200 2 x 5 - 3 x 4 - 15 x 3 + 2 x + 10, R = 0 55. 4 2 -6 -5 1 57. 4 4 -7 -6 –2 0 3 1 -6 3 6 - 1 2 4 0 -6 -2 2 - 3 2 4 -2 -4 0 4 x 3 – 6 x – 2, R = 2 (note that the terms must be reordered as shown) 4 x 2 - 2 x - 4, R = 0 59. 3 -2 2 -3 1 1.2 -0.32 0.672 -0.9312 0.4 3 -0.8 1.68 -2.328 0.0688 3 x 3 - 0.8 x 2 + 1.68 x - 2.328, R = 0.0688 61. 3 2 5 0 -7 -3 -2.4 0.32 -4.256 3.4048 2.87616 -0.8 3 -0.4 5.32 -4.256 -3.5952 -0.12384 3 x 4 - 0.4 x 3 + 5.32 x 2 - 4.256 x - 3.5952, R = -0.12384 63. ( A) P ( x ) = x 3 – 3 x 2 n = 3 a n = 1 Form a synthetic division table; the last line of each division is shown.
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