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 $\frac{7x+13}{3{x}^{2}+8x+15}=\frac{A}{x+1}+\frac{B}{3x+5}$, we eventually simplify to $7x+13=A\left(3x+5\right)+B\left(x+1\right)$. 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. $\frac{5x+16}{{x}^{2}+10x+24}$

7. $\frac{3x - 79}{{x}^{2}-5x - 24}$

8. $\frac{-x - 24}{{x}^{2}-2x - 24}$

9. $\frac{10x+47}{{x}^{2}+7x+10}$

10. $\frac{x}{6{x}^{2}+25x+25}$

11. $\frac{32x - 11}{20{x}^{2}-13x+2}$

12. $\frac{x+1}{{x}^{2}+7x+10}$

13. $\frac{5x}{{x}^{2}-9}$

14. $\frac{10x}{{x}^{2}-25}$

15. $\frac{6x}{{x}^{2}-4}$

16. $\frac{2x - 3}{{x}^{2}-6x+5}$

17. $\frac{4x - 1}{{x}^{2}-x - 6}$

18. $\frac{4x+3}{{x}^{2}+8x+15}$

19. $\frac{3x - 1}{{x}^{2}-5x+6}$

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

20. $\frac{-5x - 19}{{\left(x+4\right)}^{2}}$

21. $\frac{x}{{\left(x - 2\right)}^{2}}$

22. $\frac{7x+14}{{\left(x+3\right)}^{2}}$

23. $\frac{-24x - 27}{{\left(4x+5\right)}^{2}}$

24. $\frac{-24x - 27}{{\left(6x - 7\right)}^{2}}$

25. $\frac{5-x}{{\left(x - 7\right)}^{2}}$

26. $\frac{5x+14}{2{x}^{2}+12x+18}$

27. $\frac{5{x}^{2}+20x+8}{2x{\left(x+1\right)}^{2}}$

28. $\frac{4{x}^{2}+55x+25}{5x{\left(3x+5\right)}^{2}}$

29. $\frac{54{x}^{3}+127{x}^{2}+80x+16}{2{x}^{2}{\left(3x+2\right)}^{2}}$

30. $\frac{{x}^{3}-5{x}^{2}+12x+144}{{x}^{2}\left({x}^{2}+12x+36\right)}$

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

31. $\frac{4{x}^{2}+6x+11}{\left(x+2\right)\left({x}^{2}+x+3\right)}$

32. $\frac{4{x}^{2}+9x+23}{\left(x - 1\right)\left({x}^{2}+6x+11\right)}$

33. $\frac{-2{x}^{2}+10x+4}{\left(x - 1\right)\left({x}^{2}+3x+8\right)}$

34. $\frac{{x}^{2}+3x+1}{\left(x+1\right)\left({x}^{2}+5x - 2\right)}$

35. $\frac{4{x}^{2}+17x - 1}{\left(x+3\right)\left({x}^{2}+6x+1\right)}$

36. $\frac{4{x}^{2}}{\left(x+5\right)\left({x}^{2}+7x - 5\right)}$

37. $\frac{4{x}^{2}+5x+3}{{x}^{3}-1}$

38. $\frac{-5{x}^{2}+18x - 4}{{x}^{3}+8}$

39. $\frac{3{x}^{2}-7x+33}{{x}^{3}+27}$

40. $\frac{{x}^{2}+2x+40}{{x}^{3}-125}$

41. $\frac{4{x}^{2}+4x+12}{8{x}^{3}-27}$

42. $\frac{-50{x}^{2}+5x - 3}{125{x}^{3}-1}$

43. $\frac{-2{x}^{3}-30{x}^{2}+36x+216}{{x}^{4}+216x}$

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

44. $\frac{3{x}^{3}+2{x}^{2}+14x+15}{{\left({x}^{2}+4\right)}^{2}}$

45. $\frac{{x}^{3}+6{x}^{2}+5x+9}{{\left({x}^{2}+1\right)}^{2}}$

46. $\frac{{x}^{3}-{x}^{2}+x - 1}{{\left({x}^{2}-3\right)}^{2}}$

47. $\frac{{x}^{2}+5x+5}{{\left(x+2\right)}^{2}}$

48. $\frac{{x}^{3}+2{x}^{2}+4x}{{\left({x}^{2}+2x+9\right)}^{2}}$

49. $\frac{{x}^{2}+25}{{\left({x}^{2}+3x+25\right)}^{2}}$

50. $\frac{2{x}^{3}+11x+7x+70}{{\left(2{x}^{2}+x+14\right)}^{2}}$

51. $\frac{5x+2}{x{\left({x}^{2}+4\right)}^{2}}$

52. $\frac{{x}^{4}+{x}^{3}+8{x}^{2}+6x+36}{x{\left({x}^{2}+6\right)}^{2}}$

53. $\frac{2x - 9}{{\left({x}^{2}-x\right)}^{2}}$

54. $\frac{5{x}^{3}-2x+1}{{\left({x}^{2}+2x\right)}^{2}}$

For the following exercises, find the partial fraction expansion.

55. $\frac{{x}^{2}+4}{{\left(x+1\right)}^{3}}$

56. $\frac{{x}^{3}-4{x}^{2}+5x+4}{{\left(x - 2\right)}^{3}}$

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

57. $\frac{7}{x+8}+\frac{5}{x - 2}-\frac{x - 1}{{x}^{2}-6x - 16}$

58. $\frac{1}{x - 4}-\frac{3}{x+6}-\frac{2x+7}{{x}^{2}+2x - 24}$

59. $\frac{2x}{{x}^{2}-16}-\frac{1 - 2x}{{x}^{2}+6x+8}-\frac{x - 5}{{x}^{2}-4x}$