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

7x+133x2+8x+15=Ax+1+B3x+5\frac{7x+13}{3{x}^{2}+8x+15}=\frac{A}{x+1}+\frac{B}{3x+5}
, we eventually simplify to
7x+13=A(3x+5)+B(x+1)7x+13=A\left(3x+5\right)+B\left(x+1\right)
. Explain how you could intelligently choose an
xx
-value that will eliminate either
AA
or
BB
and solve for
AA
and
BB
.

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.

6.
5x+16x2+10x+24\frac{5x+16}{{x}^{2}+10x+24}


7.
3x79x25x24\frac{3x - 79}{{x}^{2}-5x - 24}


8. 
x24x22x24\frac{-x - 24}{{x}^{2}-2x - 24}


9.
10x+47x2+7x+10\frac{10x+47}{{x}^{2}+7x+10}


10. 
x6x2+25x+25\frac{x}{6{x}^{2}+25x+25}


11.
32x1120x213x+2\frac{32x - 11}{20{x}^{2}-13x+2}


12. 
x+1x2+7x+10\frac{x+1}{{x}^{2}+7x+10}


13.
5xx29\frac{5x}{{x}^{2}-9}


14. 
10xx225\frac{10x}{{x}^{2}-25}


15.
6xx24\frac{6x}{{x}^{2}-4}


16. 
2x3x26x+5\frac{2x - 3}{{x}^{2}-6x+5}


17.
4x1x2x6\frac{4x - 1}{{x}^{2}-x - 6}


18. 
4x+3x2+8x+15\frac{4x+3}{{x}^{2}+8x+15}


19.
3x1x25x+6\frac{3x - 1}{{x}^{2}-5x+6}


For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.

20.
5x19(x+4)2\frac{-5x - 19}{{\left(x+4\right)}^{2}}


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


22. 
7x+14(x+3)2\frac{7x+14}{{\left(x+3\right)}^{2}}


23.
24x27(4x+5)2\frac{-24x - 27}{{\left(4x+5\right)}^{2}}


24. 
24x27(6x7)2\frac{-24x - 27}{{\left(6x - 7\right)}^{2}}


25.
5x(x7)2\frac{5-x}{{\left(x - 7\right)}^{2}}


26. 
5x+142x2+12x+18\frac{5x+14}{2{x}^{2}+12x+18}


27.
5x2+20x+82x(x+1)2\frac{5{x}^{2}+20x+8}{2x{\left(x+1\right)}^{2}}


28. 
4x2+55x+255x(3x+5)2\frac{4{x}^{2}+55x+25}{5x{\left(3x+5\right)}^{2}}


29.
54x3+127x2+80x+162x2(3x+2)2\frac{54{x}^{3}+127{x}^{2}+80x+16}{2{x}^{2}{\left(3x+2\right)}^{2}}


30. 
x35x2+12x+144x2(x2+12x+36)\frac{{x}^{3}-5{x}^{2}+12x+144}{{x}^{2}\left({x}^{2}+12x+36\right)}


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)\frac{4{x}^{2}+6x+11}{\left(x+2\right)\left({x}^{2}+x+3\right)}


32. 
4x2+9x+23(x1)(x2+6x+11)\frac{4{x}^{2}+9x+23}{\left(x - 1\right)\left({x}^{2}+6x+11\right)}


33.
2x2+10x+4(x1)(x2+3x+8)\frac{-2{x}^{2}+10x+4}{\left(x - 1\right)\left({x}^{2}+3x+8\right)}


34. 
x2+3x+1(x+1)(x2+5x2)\frac{{x}^{2}+3x+1}{\left(x+1\right)\left({x}^{2}+5x - 2\right)}


35.
4x2+17x1(x+3)(x2+6x+1)\frac{4{x}^{2}+17x - 1}{\left(x+3\right)\left({x}^{2}+6x+1\right)}


36. 
4x2(x+5)(x2+7x5)\frac{4{x}^{2}}{\left(x+5\right)\left({x}^{2}+7x - 5\right)}


37.
4x2+5x+3x31\frac{4{x}^{2}+5x+3}{{x}^{3}-1}


38. 
5x2+18x4x3+8\frac{-5{x}^{2}+18x - 4}{{x}^{3}+8}


39.
3x27x+33x3+27\frac{3{x}^{2}-7x+33}{{x}^{3}+27}


40. 
x2+2x+40x3125\frac{{x}^{2}+2x+40}{{x}^{3}-125}


41.
4x2+4x+128x327\frac{4{x}^{2}+4x+12}{8{x}^{3}-27}


42. 
50x2+5x3125x31\frac{-50{x}^{2}+5x - 3}{125{x}^{3}-1}


43.
2x330x2+36x+216x4+216x\frac{-2{x}^{3}-30{x}^{2}+36x+216}{{x}^{4}+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\frac{3{x}^{3}+2{x}^{2}+14x+15}{{\left({x}^{2}+4\right)}^{2}}


45.
x3+6x2+5x+9(x2+1)2\frac{{x}^{3}+6{x}^{2}+5x+9}{{\left({x}^{2}+1\right)}^{2}}


46. 
x3x2+x1(x23)2\frac{{x}^{3}-{x}^{2}+x - 1}{{\left({x}^{2}-3\right)}^{2}}


47.
x2+5x+5(x+2)2\frac{{x}^{2}+5x+5}{{\left(x+2\right)}^{2}}


48. 
x3+2x2+4x(x2+2x+9)2\frac{{x}^{3}+2{x}^{2}+4x}{{\left({x}^{2}+2x+9\right)}^{2}}


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


50. 
2x3+11x+7x+70(2x2+x+14)2\frac{2{x}^{3}+11x+7x+70}{{\left(2{x}^{2}+x+14\right)}^{2}}


51.
5x+2x(x2+4)2\frac{5x+2}{x{\left({x}^{2}+4\right)}^{2}}


52. 
x4+x3+8x2+6x+36x(x2+6)2\frac{{x}^{4}+{x}^{3}+8{x}^{2}+6x+36}{x{\left({x}^{2}+6\right)}^{2}}


53.
2x9(x2x)2\frac{2x - 9}{{\left({x}^{2}-x\right)}^{2}}


54. 
5x32x+1(x2+2x)2\frac{5{x}^{3}-2x+1}{{\left({x}^{2}+2x\right)}^{2}}


For the following exercises, find the partial fraction expansion.

55.
x2+4(x+1)3\frac{{x}^{2}+4}{{\left(x+1\right)}^{3}}


56. 
x34x2+5x+4(x2)3\frac{{x}^{3}-4{x}^{2}+5x+4}{{\left(x - 2\right)}^{3}}


For the following exercises, perform the operation and then find the partial fraction decomposition.

57.
7x+8+5x2x1x26x16\frac{7}{x+8}+\frac{5}{x - 2}-\frac{x - 1}{{x}^{2}-6x - 16}


58. 
1x43x+62x+7x2+2x24\frac{1}{x - 4}-\frac{3}{x+6}-\frac{2x+7}{{x}^{2}+2x - 24}


59.
2xx21612xx2+6x+8x5x24x\frac{2x}{{x}^{2}-16}-\frac{1 - 2x}{{x}^{2}+6x+8}-\frac{x - 5}{{x}^{2}-4x}

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