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. 3x−79x2−5x−24
8. −x−24x2−2x−24
9. 10x+47x2+7x+10
10. x6x2+25x+25
11. 32x−1120x2−13x+2
12. x+1x2+7x+10
13. 5xx2−9
14. 10xx2−25
15. 6xx2−4
16. 2x−3x2−6x+5
17. 4x−1x2−x−6
18. 4x+3x2+8x+15
19. 3x−1x2−5x+6
For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.
20. −5x−19(x+4)2
21. x(x−2)2
22. 7x+14(x+3)2
23. −24x−27(4x+5)2
24. −24x−27(6x−7)2
25. 5−x(x−7)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. x3−5x2+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(x−1)(x2+6x+11)
33. −2x2+10x+4(x−1)(x2+3x+8)
34. x2+3x+1(x+1)(x2+5x−2)
35. 4x2+17x−1(x+3)(x2+6x+1)
36. 4x2(x+5)(x2+7x−5)
37. 4x2+5x+3x3−1
38. −5x2+18x−4x3+8
39. 3x2−7x+33x3+27
40. x2+2x+40x3−125
41. 4x2+4x+128x3−27
42. −50x2+5x−3125x3−1
43. −2x3−30x2+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. x3−x2+x−1(x2−3)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. 2x−9(x2−x)2
54. 5x3−2x+1(x2+2x)2
For the following exercises, find the partial fraction expansion.
55. x2+4(x+1)3
56. x3−4x2+5x+4(x−2)3
For the following exercises, perform the operation and then find the partial fraction decomposition.
57. 7x+8+5x−2−x−1x2−6x−16
58. 1x−4−3x+6−2x+7x2+2x−24
59. 2xx2−16−1−2xx2+6x+8−x−5x2−4x
Candela Citations
- Precalculus. Authored by: Jay Abramson, et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution. License Terms: Download for free at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface