Section Exercises

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 [latex]\frac{7x+13}{3{x}^{2}+8x+15}=\frac{A}{x+1}+\frac{B}{3x+5}[/latex], we eventually simplify to [latex]7x+13=A\left(3x+5\right)+B\left(x+1\right)[/latex]. Explain how you could intelligently choose an [latex]x[/latex] -value that will eliminate either [latex]A[/latex] or [latex]B[/latex] and solve for [latex]A[/latex] and [latex]B[/latex]. For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors. 6. [latex]\frac{5x+16}{{x}^{2}+10x+24}[/latex] 7. [latex]\frac{3x - 79}{{x}^{2}-5x - 24}[/latex] 8. [latex]\frac{-x - 24}{{x}^{2}-2x - 24}[/latex] 9. [latex]\frac{10x+47}{{x}^{2}+7x+10}[/latex] 10. [latex]\frac{x}{6{x}^{2}+25x+25}[/latex] 11. [latex]\frac{32x - 11}{20{x}^{2}-13x+2}[/latex] 12. [latex]\frac{x+1}{{x}^{2}+7x+10}[/latex] 13. [latex]\frac{5x}{{x}^{2}-9}[/latex] 14. [latex]\frac{10x}{{x}^{2}-25}[/latex] 15. [latex]\frac{6x}{{x}^{2}-4}[/latex] 16. [latex]\frac{2x - 3}{{x}^{2}-6x+5}[/latex] 17. [latex]\frac{4x - 1}{{x}^{2}-x - 6}[/latex] 18. [latex]\frac{4x+3}{{x}^{2}+8x+15}[/latex] 19. [latex]\frac{3x - 1}{{x}^{2}-5x+6}[/latex] For the following exercises, find the decomposition of the partial fraction for the repeating linear factors. 20. [latex]\frac{-5x - 19}{{\left(x+4\right)}^{2}}[/latex] 21. [latex]\frac{x}{{\left(x - 2\right)}^{2}}[/latex] 22. [latex]\frac{7x+14}{{\left(x+3\right)}^{2}}[/latex] 23. [latex]\frac{-24x - 27}{{\left(4x+5\right)}^{2}}[/latex] 24. [latex]\frac{-24x - 27}{{\left(6x - 7\right)}^{2}}[/latex] 25. [latex]\frac{5-x}{{\left(x - 7\right)}^{2}}[/latex] 26. [latex]\frac{5x+14}{2{x}^{2}+12x+18}[/latex] 27. [latex]\frac{5{x}^{2}+20x+8}{2x{\left(x+1\right)}^{2}}[/latex] 28. [latex]\frac{4{x}^{2}+55x+25}{5x{\left(3x+5\right)}^{2}}[/latex] 29. [latex]\frac{54{x}^{3}+127{x}^{2}+80x+16}{2{x}^{2}{\left(3x+2\right)}^{2}}[/latex] 30. [latex]\frac{{x}^{3}-5{x}^{2}+12x+144}{{x}^{2}\left({x}^{2}+12x+36\right)}[/latex] For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. 31. [latex]\frac{4{x}^{2}+6x+11}{\left(x+2\right)\left({x}^{2}+x+3\right)}[/latex] 32. [latex]\frac{4{x}^{2}+9x+23}{\left(x - 1\right)\left({x}^{2}+6x+11\right)}[/latex] 33. [latex]\frac{-2{x}^{2}+10x+4}{\left(x - 1\right)\left({x}^{2}+3x+8\right)}[/latex] 34. [latex]\frac{{x}^{2}+3x+1}{\left(x+1\right)\left({x}^{2}+5x - 2\right)}[/latex] 35. [latex]\frac{4{x}^{2}+17x - 1}{\left(x+3\right)\left({x}^{2}+6x+1\right)}[/latex] 36. [latex]\frac{4{x}^{2}}{\left(x+5\right)\left({x}^{2}+7x - 5\right)}[/latex] 37. [latex]\frac{4{x}^{2}+5x+3}{{x}^{3}-1}[/latex] 38. [latex]\frac{-5{x}^{2}+18x - 4}{{x}^{3}+8}[/latex] 39. [latex]\frac{3{x}^{2}-7x+33}{{x}^{3}+27}[/latex] 40. [latex]\frac{{x}^{2}+2x+40}{{x}^{3}-125}[/latex] 41. [latex]\frac{4{x}^{2}+4x+12}{8{x}^{3}-27}[/latex] 42. [latex]\frac{-50{x}^{2}+5x - 3}{125{x}^{3}-1}[/latex] 43. [latex]\frac{-2{x}^{3}-30{x}^{2}+36x+216}{{x}^{4}+216x}[/latex] For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor. 44. [latex]\frac{3{x}^{3}+2{x}^{2}+14x+15}{{\left({x}^{2}+4\right)}^{2}}[/latex] 45. [latex]\frac{{x}^{3}+6{x}^{2}+5x+9}{{\left({x}^{2}+1\right)}^{2}}[/latex] 46. [latex]\frac{{x}^{3}-{x}^{2}+x - 1}{{\left({x}^{2}-3\right)}^{2}}[/latex] 47. [latex]\frac{{x}^{2}+5x+5}{{\left(x+2\right)}^{2}}[/latex] 48. [latex]\frac{{x}^{3}+2{x}^{2}+4x}{{\left({x}^{2}+2x+9\right)}^{2}}[/latex] 49. [latex]\frac{{x}^{2}+25}{{\left({x}^{2}+3x+25\right)}^{2}}[/latex] 50. [latex]\frac{2{x}^{3}+11x+7x+70}{{\left(2{x}^{2}+x+14\right)}^{2}}[/latex] 51. [latex]\frac{5x+2}{x{\left({x}^{2}+4\right)}^{2}}[/latex] 52. [latex]\frac{{x}^{4}+{x}^{3}+8{x}^{2}+6x+36}{x{\left({x}^{2}+6\right)}^{2}}[/latex] 53. [latex]\frac{2x - 9}{{\left({x}^{2}-x\right)}^{2}}[/latex] 54. [latex]\frac{5{x}^{3}-2x+1}{{\left({x}^{2}+2x\right)}^{2}}[/latex] For the following exercises, find the partial fraction expansion. 55. [latex]\frac{{x}^{2}+4}{{\left(x+1\right)}^{3}}[/latex] 56. [latex]\frac{{x}^{3}-4{x}^{2}+5x+4}{{\left(x - 2\right)}^{3}}[/latex] For the following exercises, perform the operation and then find the partial fraction decomposition. 57. [latex]\frac{7}{x+8}+\frac{5}{x - 2}-\frac{x - 1}{{x}^{2}-6x - 16}[/latex] 58. [latex]\frac{1}{x - 4}-\frac{3}{x+6}-\frac{2x+7}{{x}^{2}+2x - 24}[/latex] 59. [latex]\frac{2x}{{x}^{2}-16}-\frac{1 - 2x}{{x}^{2}+6x+8}-\frac{x - 5}{{x}^{2}-4x}[/latex]

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