820 CHAPTER 11 Systems of Equations and Inequalities In Problems 17–50, find the partial fraction decomposition of each rational expression. 17. ( ) − x x 4 1 18. ( )( ) + − x x x 3 2 1 19. ( ) + x x 1 1 2 20. ( ) ( ) + + x x 1 1 4 2 21. ( )( ) − − x x x 1 2 22. ( )( ) + − x x x 3 2 4 23. ( ) ( ) − + x x x 1 1 2 2 24. ( ) + − x x x 1 2 2 25. − x 1 8 3 26. + − x x 2 4 1 3 27. ( ) ( ) − + x x x 1 1 2 2 2 28. ( ) + − x x x 1 2 2 2 29. ( )( ) − + + x x x 3 2 1 2 30. ( )( ) + + − x x x x 2 1 2 2 31. ( ) + + x x x 4 4 2 2 32. ( ) ( ) + − + x x x x 10 2 1 2 2 2 2 33. ( ) ( ) + + + + + x x x x x 2 3 1 2 4 2 2 34. ( ) − − + + x x x x x 11 18 3 3 2 2 35. ( )( ) − + x x x 3 2 2 1 36. ( )( ) + − x x 1 2 3 4 1 37. + − x x x2 3 2 38. ( ) ( ) − − + + + x x x x x 8 1 5 6 2 2 39. ( ) + + + x x x 2 3 4 2 2 2 40. ( ) + + x x 1 16 3 2 2 41. + − − x x x x 7 3 2 3 3 2 42. + − x x x 1 3 5 4 43. − + − x x x x 4 5 2 2 3 2 44. + + − + x x x x 1 5 3 2 3 2 45. ( ) + x x 16 3 2 3 46. ( ) + x x 4 2 2 3 47. − − x x 4 2 5 3 2 48. + − x x x 4 2 3 2 2 49. + − x x x 2 3 9 4 2 50. + − − x x x 9 2 8 2 4 2 Mixed Practice In Problems 51–58, use the division algorithm to rewrite each improper rational expression as the sum of a polynomial and a proper rational expression. Find the partial fraction decomposition of the proper rational expression. Finally, express the improper rational expression as the sum of a polynomial and the partial fraction decomposition. 51. + − + − x x x x 3 3 4 3 2 2 52. − + + + x x x x 3 1 5 6 3 2 2 53. + x x 1 3 2 54. + + x x x 4 3 2 55. − + − + + x x x x x 5 4 4 4 4 2 2 56. + − + − + x x x x x 2 2 1 4 3 2 57. + − + − + x x x x x 2 2 1 5 4 2 4 2 58. − + + + + x x x x x 1 6 9 5 3 2 4 2 59. Challenge Problem Use a substitution and partial fraction decomposition to express + − e e e 3 2 x x x 2 in terms of e .x 60. Challenge Problem Use a substitution and partial fraction decomposition to express −x x 2 3 in terms of x. 3 Retain Your Knowledge Problems 61–70 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 61. Credit Card Balance Nick has a credit card balance of $4200. If the credit card company charges 18% interest compound daily, and Nick does not make any payments on the account, how long will it take for his balance to double? Round to two decimal places. 62. If ( ) = + f x x 4 and ( ) = − g x x x3 , 2 find ( )( ) − g f 3 . 63. Find the exact value of ° ° sec 52 cos 308 . 64. Plot the point given by the polar coordinates π ( ) −1, 5 4 and find its rectangular coordinates. 65. Determine whether ( ) = − − f x x x 3 10 2 is even, odd, or neither. 66. The function ( ) = − − f x 8 4 x 3 is one-to-one. Find −f .1 67. Find an equation for the hyperbola with vertices ( ) − 0, 5 and ( ) 0, 5 , and a focus at ( ) 0, 13 . 68. Solve: − ≥ + x x 5 2 1 4 5 69. Solve for − − + = D x xD y yD D : 2 4 4 2 70. The normal line is the line that is perpendicular to the tangent line at the point of tangency. If = − + y x2 2 is the tangent line to ( ) = − + f x x x 1 3 4 5, 2 find an equation of the normal line to f at the point of tangency. ‘Are You Prepared?’ Answers 1. True 2. True 3. +x 3 4 4. True
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