Solutions to Try Its

1. $\frac{3}{x - 3}-\frac{2}{x - 2}$

2. $\frac{6}{x - 1}-\frac{5}{{\left(x - 1\right)}^{2}}$

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

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

Solutions to Odd-Numbered Exercises

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

3. Graph both sides and ensure they are equal.

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

7. $\frac{8}{x+3}-\frac{5}{x - 8}$

9. $\frac{1}{x+5}+\frac{9}{x+2}$

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

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

15. $\frac{3}{x+2}+\frac{3}{x - 2}$

17. $\frac{9}{5\left(x+2\right)}+\frac{11}{5\left(x - 3\right)}$

19. $\frac{8}{x - 3}-\frac{5}{x - 2}$

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

23. $-\frac{6}{4x+5}+\frac{3}{{\left(4x+5\right)}^{2}}$

25. $-\frac{1}{x - 7}-\frac{2}{{\left(x - 7\right)}^{2}}$

27. $\frac{4}{x}-\frac{3}{2\left(x+1\right)}+\frac{7}{2{\left(x+1\right)}^{2}}$

29. $\frac{4}{x}+\frac{2}{{x}^{2}}-\frac{3}{3x+2}+\frac{7}{2{\left(3x+2\right)}^{2}}$

31. $\frac{x+1}{{x}^{2}+x+3}+\frac{3}{x+2}$

33. $\frac{4 - 3x}{{x}^{2}+3x+8}+\frac{1}{x - 1}$

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

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

39. $\frac{2}{{x}^{2}-3x+9}+\frac{3}{x+3}$

41. $-\frac{1}{4{x}^{2}+6x+9}+\frac{1}{2x - 3}$

43. $\frac{1}{x}+\frac{1}{x+6}-\frac{4x}{{x}^{2}-6x+36}$

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

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

49. $\frac{1}{{x}^{2}+3x+25}-\frac{3x}{{\left({x}^{2}+3x+25\right)}^{2}}$

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

53. $-\frac{16}{x}-\frac{9}{{x}^{2}}+\frac{16}{x - 1}-\frac{7}{{\left(x - 1\right)}^{2}}$

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

57. $\frac{5}{x - 2}-\frac{3}{10\left(x+2\right)}+\frac{7}{x+8}-\frac{7}{10\left(x - 8\right)}$

59. $-\frac{5}{4x}-\frac{5}{2\left(x+2\right)}+\frac{11}{2\left(x+4\right)}+\frac{5}{4\left(x+4\right)}$