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Study Guides > College Algebra

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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  • Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution.