Solutions for Partial Fractions

Solutions to Try Its

1. [latex]\frac{3}{x - 3}-\frac{2}{x - 2}[/latex]

2. [latex]\frac{6}{x - 1}-\frac{5}{{\left(x - 1\right)}^{2}}[/latex]

3. [latex]\frac{3}{x - 1}+\frac{2x - 4}{{x}^{2}+1}[/latex]

4. [latex]\frac{x - 2}{{x}^{2}-2x+3}+\frac{2x+1}{{\left({x}^{2}-2x+3\right)}^{2}}[/latex]

Solutions to Odd-Numbered Exercises

1. No, a quotient of polynomials can only be decomposed if the denominator can be factored. For example, [latex]\frac{1}{{x}^{2}+1}[/latex] cannot be decomposed because the denominator cannot be factored.

3. Graph both sides and ensure they are equal.

5. If we choose [latex]x=-1[/latex], then the B-term disappears, letting us immediately know that [latex]A=3[/latex]. We could alternatively plug in [latex]x=-\frac{5}{3}[/latex], giving us a B-value of [latex]-2[/latex].

7. [latex]\frac{8}{x+3}-\frac{5}{x - 8}[/latex]

9. [latex]\frac{1}{x+5}+\frac{9}{x+2}[/latex]

11. [latex]\frac{3}{5x - 2}+\frac{4}{4x - 1}[/latex]

13. [latex]\frac{5}{2\left(x+3\right)}+\frac{5}{2\left(x - 3\right)}[/latex]

15. [latex]\frac{3}{x+2}+\frac{3}{x - 2}[/latex]

17. [latex]\frac{9}{5\left(x+2\right)}+\frac{11}{5\left(x - 3\right)}[/latex]

19. [latex]\frac{8}{x - 3}-\frac{5}{x - 2}[/latex]

21. [latex]\frac{1}{x - 2}+\frac{2}{{\left(x - 2\right)}^{2}}[/latex]

23. [latex]-\frac{6}{4x+5}+\frac{3}{{\left(4x+5\right)}^{2}}[/latex]

25. [latex]-\frac{1}{x - 7}-\frac{2}{{\left(x - 7\right)}^{2}}[/latex]

27. [latex]\frac{4}{x}-\frac{3}{2\left(x+1\right)}+\frac{7}{2{\left(x+1\right)}^{2}}[/latex]

29. [latex]\frac{4}{x}+\frac{2}{{x}^{2}}-\frac{3}{3x+2}+\frac{7}{2{\left(3x+2\right)}^{2}}[/latex]

31. [latex]\frac{x+1}{{x}^{2}+x+3}+\frac{3}{x+2}[/latex]

33. [latex]\frac{4 - 3x}{{x}^{2}+3x+8}+\frac{1}{x - 1}[/latex]

35. [latex]\frac{2x - 1}{{x}^{2}+6x+1}+\frac{2}{x+3}[/latex]

37. [latex]\frac{1}{{x}^{2}+x+1}+\frac{4}{x - 1}[/latex]

39. [latex]\frac{2}{{x}^{2}-3x+9}+\frac{3}{x+3}[/latex]

41. [latex]-\frac{1}{4{x}^{2}+6x+9}+\frac{1}{2x - 3}[/latex]

43. [latex]\frac{1}{x}+\frac{1}{x+6}-\frac{4x}{{x}^{2}-6x+36}[/latex]

45. [latex]\frac{x+6}{{x}^{2}+1}+\frac{4x+3}{{\left({x}^{2}+1\right)}^{2}}[/latex]

47. [latex]\frac{x+1}{x+2}+\frac{2x+3}{{\left(x+2\right)}^{2}}[/latex]

49. [latex]\frac{1}{{x}^{2}+3x+25}-\frac{3x}{{\left({x}^{2}+3x+25\right)}^{2}}[/latex]

51. [latex]\frac{1}{8x}-\frac{x}{8\left({x}^{2}+4\right)}+\frac{10-x}{2{\left({x}^{2}+4\right)}^{2}}[/latex]

53. [latex]-\frac{16}{x}-\frac{9}{{x}^{2}}+\frac{16}{x - 1}-\frac{7}{{\left(x - 1\right)}^{2}}[/latex]

55. [latex]\frac{1}{x+1}-\frac{2}{{\left(x+1\right)}^{2}}+\frac{5}{{\left(x+1\right)}^{3}}[/latex]

57. [latex]\frac{5}{x - 2}-\frac{3}{10\left(x+2\right)}+\frac{7}{x+8}-\frac{7}{10\left(x - 8\right)}[/latex]

59. [latex]-\frac{5}{4x}-\frac{5}{2\left(x+2\right)}+\frac{11}{2\left(x+4\right)}+\frac{5}{4\left(x+4\right)}[/latex]