Solutions

Solutions to Try Its

1. 
3x32x2\frac{3}{x - 3}-\frac{2}{x - 2}


2. 
6x15(x1)2\frac{6}{x - 1}-\frac{5}{{\left(x - 1\right)}^{2}}


3. 
3x1+2x4x2+1\frac{3}{x - 1}+\frac{2x - 4}{{x}^{2}+1}


4. 
x2x22x+3+2x+1(x22x+3)2\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,
1x2+1\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=1x=-1
, then the B-term disappears, letting us immediately know that
A=3A=3
. We could alternatively plug in
x=53x=-\frac{5}{3}
, giving us a B-value of
2-2
.

7. 
8x+35x8\frac{8}{x+3}-\frac{5}{x - 8}


9. 
1x+5+9x+2\frac{1}{x+5}+\frac{9}{x+2}


11. 
35x2+44x1\frac{3}{5x - 2}+\frac{4}{4x - 1}


13. 
52(x+3)+52(x3)\frac{5}{2\left(x+3\right)}+\frac{5}{2\left(x - 3\right)}


15. 
3x+2+3x2\frac{3}{x+2}+\frac{3}{x - 2}


17. 
95(x+2)+115(x3)\frac{9}{5\left(x+2\right)}+\frac{11}{5\left(x - 3\right)}


19. 
8x35x2\frac{8}{x - 3}-\frac{5}{x - 2}


21. 
1x2+2(x2)2\frac{1}{x - 2}+\frac{2}{{\left(x - 2\right)}^{2}}


23. 
64x+5+3(4x+5)2-\frac{6}{4x+5}+\frac{3}{{\left(4x+5\right)}^{2}}


25. 
1x72(x7)2-\frac{1}{x - 7}-\frac{2}{{\left(x - 7\right)}^{2}}


27. 
4x32(x+1)+72(x+1)2\frac{4}{x}-\frac{3}{2\left(x+1\right)}+\frac{7}{2{\left(x+1\right)}^{2}}


29. 
4x+2x233x+2+72(3x+2)2\frac{4}{x}+\frac{2}{{x}^{2}}-\frac{3}{3x+2}+\frac{7}{2{\left(3x+2\right)}^{2}}


31. 
x+1x2+x+3+3x+2\frac{x+1}{{x}^{2}+x+3}+\frac{3}{x+2}


33. 
43xx2+3x+8+1x1\frac{4 - 3x}{{x}^{2}+3x+8}+\frac{1}{x - 1}


35. 
2x1x2+6x+1+2x+3\frac{2x - 1}{{x}^{2}+6x+1}+\frac{2}{x+3}


37. 
1x2+x+1+4x1\frac{1}{{x}^{2}+x+1}+\frac{4}{x - 1}


39. 
2x23x+9+3x+3\frac{2}{{x}^{2}-3x+9}+\frac{3}{x+3}


41. 
14x2+6x+9+12x3-\frac{1}{4{x}^{2}+6x+9}+\frac{1}{2x - 3}


43. 
1x+1x+64xx26x+36\frac{1}{x}+\frac{1}{x+6}-\frac{4x}{{x}^{2}-6x+36}


45. 
x+6x2+1+4x+3(x2+1)2\frac{x+6}{{x}^{2}+1}+\frac{4x+3}{{\left({x}^{2}+1\right)}^{2}}


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


49. 
1x2+3x+253x(x2+3x+25)2\frac{1}{{x}^{2}+3x+25}-\frac{3x}{{\left({x}^{2}+3x+25\right)}^{2}}


51. 
18xx8(x2+4)+10x2(x2+4)2\frac{1}{8x}-\frac{x}{8\left({x}^{2}+4\right)}+\frac{10-x}{2{\left({x}^{2}+4\right)}^{2}}


53. 
16x9x2+16x17(x1)2-\frac{16}{x}-\frac{9}{{x}^{2}}+\frac{16}{x - 1}-\frac{7}{{\left(x - 1\right)}^{2}}


55. 
1x+12(x+1)2+5(x+1)3\frac{1}{x+1}-\frac{2}{{\left(x+1\right)}^{2}}+\frac{5}{{\left(x+1\right)}^{3}}


57. 
5x2310(x+2)+7x+8710(x8)\frac{5}{x - 2}-\frac{3}{10\left(x+2\right)}+\frac{7}{x+8}-\frac{7}{10\left(x - 8\right)}


59. 
54x52(x+2)+112(x+4)+54(x+4)-\frac{5}{4x}-\frac{5}{2\left(x+2\right)}+\frac{11}{2\left(x+4\right)}+\frac{5}{4\left(x+4\right)}


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