Problem Set 32: Partial Fractions

1. Can any quotient of polynomials be decomposed into at least two partial fractions? If so, explain why, and if not, give an example of such a fraction

2. Can you explain why a partial fraction decomposition is unique? (Hint: Think about it as a system of equations.)

3. Can you explain how to verify a partial fraction decomposition graphically?

4. You are unsure if you correctly decomposed the partial fraction correctly. Explain how you could double-check your answer.

5. Once you have a system of equations generated by the partial fraction decomposition, can you explain another method to solve it? For example if you had 7x+133x2+8x+15=Ax+1+B3x+5, we eventually simplify to 7x+13=A(3x+5)+B(x+1). Explain how you could intelligently choose an x -value that will eliminate either A or B and solve for A and B.

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.

6. 5x+16x2+10x+24

7. 3x79x25x24

8. x24x22x24

9. 10x+47x2+7x+10

10. x6x2+25x+25

11. 32x1120x213x+2

12. x+1x2+7x+10

13. 5xx29

14. 10xx225

15. 6xx24

16. 2x3x26x+5

17. 4x1x2x6

18. 4x+3x2+8x+15

19. 3x1x25x+6

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.

20. 5x19(x+4)2

21. x(x2)2

22. 7x+14(x+3)2

23. 24x27(4x+5)2

24. 24x27(6x7)2

25. 5x(x7)2

26. 5x+142x2+12x+18

27. 5x2+20x+82x(x+1)2

28. 4x2+55x+255x(3x+5)2

29. 54x3+127x2+80x+162x2(3x+2)2

30. x35x2+12x+144x2(x2+12x+36)

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

31. 4x2+6x+11(x+2)(x2+x+3)

32. 4x2+9x+23(x1)(x2+6x+11)

33. 2x2+10x+4(x1)(x2+3x+8)

34. x2+3x+1(x+1)(x2+5x2)

35. 4x2+17x1(x+3)(x2+6x+1)

36. 4x2(x+5)(x2+7x5)

37. 4x2+5x+3x31

38. 5x2+18x4x3+8

39. 3x27x+33x3+27

40. x2+2x+40x3125

41. 4x2+4x+128x327

42. 50x2+5x3125x31

43. 2x330x2+36x+216x4+216x

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.

44. 3x3+2x2+14x+15(x2+4)2

45. x3+6x2+5x+9(x2+1)2

46. x3x2+x1(x23)2

47. x2+5x+5(x+2)2

48. x3+2x2+4x(x2+2x+9)2

49. x2+25(x2+3x+25)2

50. 2x3+11x+7x+70(2x2+x+14)2

51. 5x+2x(x2+4)2

52. x4+x3+8x2+6x+36x(x2+6)2

53. 2x9(x2x)2

54. 5x32x+1(x2+2x)2

For the following exercises, find the partial fraction expansion.

55. x2+4(x+1)3

56. x34x2+5x+4(x2)3

For the following exercises, perform the operation and then find the partial fraction decomposition.

57. 7x+8+5x2x1x26x16

58. 1x43x+62x+7x2+2x24

59. 2xx21612xx2+6x+8x5x24x