{"id":15608,"date":"2019-09-04T23:55:09","date_gmt":"2019-09-04T23:55:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&p=15608"},"modified":"2025-02-05T05:19:55","modified_gmt":"2025-02-05T05:19:55","slug":"problem-set-32-partial-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/problem-set-32-partial-fractions\/","title":{"raw":"Problem Set 32: Partial Fractions","rendered":"Problem Set 32: Partial Fractions"},"content":{"raw":"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\r\n\r\n2.\u00a0Can you explain why a partial fraction decomposition is unique? (Hint: Think about it as a system of equations.)\r\n\r\n3. Can you explain how to verify a partial fraction decomposition graphically?\r\n\r\n4.\u00a0You are unsure if you correctly decomposed the partial fraction correctly. Explain how you could double-check your answer.\r\n\r\n5. 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 [latex]\\frac{7x+13}{3{x}^{2}+8x+15}=\\frac{A}{x+1}+\\frac{B}{3x+5}[\/latex], we eventually simplify to [latex]7x+13=A\\left(3x+5\\right)+B\\left(x+1\\right)[\/latex]. Explain how you could intelligently choose an [latex]x[\/latex] -value that will eliminate either [latex]A[\/latex] or [latex]B[\/latex] and solve for [latex]A[\/latex] and [latex]B[\/latex].\r\n\r\nFor the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.\r\n\r\n6. [latex]\\frac{5x+16}{{x}^{2}+10x+24}[\/latex]\r\n\r\n7. [latex]\\frac{3x - 79}{{x}^{2}-5x - 24}[\/latex]\r\n\r\n8.\u00a0[latex]\\frac{-x - 24}{{x}^{2}-2x - 24}[\/latex]\r\n\r\n9. [latex]\\frac{10x+47}{{x}^{2}+7x+10}[\/latex]\r\n\r\n10.\u00a0[latex]\\frac{x}{6{x}^{2}+25x+25}[\/latex]\r\n\r\n11. [latex]\\frac{32x - 11}{20{x}^{2}-13x+2}[\/latex]\r\n\r\n12.\u00a0[latex]\\frac{x+1}{{x}^{2}+7x+10}[\/latex]\r\n\r\n13. [latex]\\frac{5x}{{x}^{2}-9}[\/latex]\r\n\r\n14.\u00a0[latex]\\frac{10x}{{x}^{2}-25}[\/latex]\r\n\r\n15. [latex]\\frac{6x}{{x}^{2}-4}[\/latex]\r\n\r\n16.\u00a0[latex]\\frac{2x - 3}{{x}^{2}-6x+5}[\/latex]\r\n\r\n17. [latex]\\frac{4x - 1}{{x}^{2}-x - 6}[\/latex]\r\n\r\n18.\u00a0[latex]\\frac{4x+3}{{x}^{2}+8x+15}[\/latex]\r\n\r\n19. [latex]\\frac{3x - 1}{{x}^{2}-5x+6}[\/latex]\r\n\r\nFor the following exercises, find the decomposition of the partial fraction for the repeating linear factors.\r\n\r\n20. [latex]\\frac{-5x - 19}{{\\left(x+4\\right)}^{2}}[\/latex]\r\n\r\n21. [latex]\\frac{x}{{\\left(x - 2\\right)}^{2}}[\/latex]\r\n\r\n22.\u00a0[latex]\\frac{7x+14}{{\\left(x+3\\right)}^{2}}[\/latex]\r\n\r\n23. [latex]\\frac{-24x - 27}{{\\left(4x+5\\right)}^{2}}[\/latex]\r\n\r\n24.\u00a0[latex]\\frac{-24x - 27}{{\\left(6x - 7\\right)}^{2}}[\/latex]\r\n\r\n25. [latex]\\frac{5-x}{{\\left(x - 7\\right)}^{2}}[\/latex]\r\n\r\n26.\u00a0[latex]\\frac{5x+14}{2{x}^{2}+12x+18}[\/latex]\r\n\r\n27. [latex]\\frac{5{x}^{2}+20x+8}{2x{\\left(x+1\\right)}^{2}}[\/latex]\r\n\r\n28.\u00a0[latex]\\frac{4{x}^{2}+55x+25}{5x{\\left(3x+5\\right)}^{2}}[\/latex]\r\n\r\n29. [latex]\\frac{54{x}^{3}+127{x}^{2}+80x+16}{2{x}^{2}{\\left(3x+2\\right)}^{2}}[\/latex]\r\n\r\n30.\u00a0[latex]\\frac{{x}^{3}-5{x}^{2}+12x+144}{{x}^{2}\\left({x}^{2}+12x+36\\right)}[\/latex]\r\n\r\nFor the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.\r\n\r\n31. [latex]\\frac{4{x}^{2}+6x+11}{\\left(x+2\\right)\\left({x}^{2}+x+3\\right)}[\/latex]\r\n\r\n32.\u00a0[latex]\\frac{4{x}^{2}+9x+23}{\\left(x - 1\\right)\\left({x}^{2}+6x+11\\right)}[\/latex]\r\n\r\n33. [latex]\\frac{-2{x}^{2}+10x+4}{\\left(x - 1\\right)\\left({x}^{2}+3x+8\\right)}[\/latex]\r\n\r\n34.\u00a0[latex]\\frac{{x}^{2}+3x+1}{\\left(x+1\\right)\\left({x}^{2}+5x - 2\\right)}[\/latex]\r\n\r\n35. [latex]\\frac{4{x}^{2}+17x - 1}{\\left(x+3\\right)\\left({x}^{2}+6x+1\\right)}[\/latex]\r\n\r\n36.\u00a0[latex]\\frac{4{x}^{2}}{\\left(x+5\\right)\\left({x}^{2}+7x - 5\\right)}[\/latex]\r\n\r\n37. [latex]\\frac{4{x}^{2}+5x+3}{{x}^{3}-1}[\/latex]\r\n\r\n38.\u00a0[latex]\\frac{-5{x}^{2}+18x - 4}{{x}^{3}+8}[\/latex]\r\n\r\n39. [latex]\\frac{3{x}^{2}-7x+33}{{x}^{3}+27}[\/latex]\r\n\r\n40.\u00a0[latex]\\frac{{x}^{2}+2x+40}{{x}^{3}-125}[\/latex]\r\n\r\n41. [latex]\\frac{4{x}^{2}+4x+12}{8{x}^{3}-27}[\/latex]\r\n\r\n42.\u00a0[latex]\\frac{-50{x}^{2}+5x - 3}{125{x}^{3}-1}[\/latex]\r\n\r\n43. [latex]\\frac{-2{x}^{3}-30{x}^{2}+36x+216}{{x}^{4}+216x}[\/latex]\r\n\r\nFor the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.\r\n\r\n44. [latex]\\frac{3{x}^{3}+2{x}^{2}+14x+15}{{\\left({x}^{2}+4\\right)}^{2}}[\/latex]\r\n\r\n45. [latex]\\frac{{x}^{3}+6{x}^{2}+5x+9}{{\\left({x}^{2}+1\\right)}^{2}}[\/latex]\r\n\r\n46.\u00a0[latex]\\frac{{x}^{3}-{x}^{2}+x - 1}{{\\left({x}^{2}-3\\right)}^{2}}[\/latex]\r\n\r\n47. [latex]\\frac{{x}^{2}+5x+5}{{\\left(x+2\\right)}^{2}}[\/latex]\r\n\r\n48.\u00a0[latex]\\frac{{x}^{3}+2{x}^{2}+4x}{{\\left({x}^{2}+2x+9\\right)}^{2}}[\/latex]\r\n\r\n49. [latex]\\frac{{x}^{2}+25}{{\\left({x}^{2}+3x+25\\right)}^{2}}[\/latex]\r\n\r\n50.\u00a0[latex]\\frac{2{x}^{3}+11x+7x+70}{{\\left(2{x}^{2}+x+14\\right)}^{2}}[\/latex]\r\n\r\n51. [latex]\\frac{5x+2}{x{\\left({x}^{2}+4\\right)}^{2}}[\/latex]\r\n\r\n52.\u00a0[latex]\\frac{{x}^{4}+{x}^{3}+8{x}^{2}+6x+36}{x{\\left({x}^{2}+6\\right)}^{2}}[\/latex]\r\n\r\n53. [latex]\\frac{2x - 9}{{\\left({x}^{2}-x\\right)}^{2}}[\/latex]\r\n\r\n54.\u00a0[latex]\\frac{5{x}^{3}-2x+1}{{\\left({x}^{2}+2x\\right)}^{2}}[\/latex]\r\n\r\nFor the following exercises, find the partial fraction expansion.\r\n\r\n55. [latex]\\frac{{x}^{2}+4}{{\\left(x+1\\right)}^{3}}[\/latex]\r\n\r\n56.\u00a0[latex]\\frac{{x}^{3}-4{x}^{2}+5x+4}{{\\left(x - 2\\right)}^{3}}[\/latex]\r\n\r\nFor the following exercises, perform the operation and then find the partial fraction decomposition.\r\n\r\n57. [latex]\\frac{7}{x+8}+\\frac{5}{x - 2}-\\frac{x - 1}{{x}^{2}-6x - 16}[\/latex]\r\n\r\n58.\u00a0[latex]\\frac{1}{x - 4}-\\frac{3}{x+6}-\\frac{2x+7}{{x}^{2}+2x - 24}[\/latex]\r\n\r\n59. [latex]\\frac{2x}{{x}^{2}-16}-\\frac{1 - 2x}{{x}^{2}+6x+8}-\\frac{x - 5}{{x}^{2}-4x}[\/latex]","rendered":"

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<\/p>\n

2.\u00a0Can you explain why a partial fraction decomposition is unique? (Hint: Think about it as a system of equations.)<\/p>\n

3. Can you explain how to verify a partial fraction decomposition graphically?<\/p>\n

4.\u00a0You are unsure if you correctly decomposed the partial fraction correctly. Explain how you could double-check your answer.<\/p>\n

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 [latex]\\frac{7x+13}{3{x}^{2}+8x+15}=\\frac{A}{x+1}+\\frac{B}{3x+5}[\/latex], we eventually simplify to [latex]7x+13=A\\left(3x+5\\right)+B\\left(x+1\\right)[\/latex]. Explain how you could intelligently choose an [latex]x[\/latex] -value that will eliminate either [latex]A[\/latex] or [latex]B[\/latex] and solve for [latex]A[\/latex] and [latex]B[\/latex].<\/p>\n

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.<\/p>\n

6. [latex]\\frac{5x+16}{{x}^{2}+10x+24}[\/latex]<\/p>\n

7. [latex]\\frac{3x - 79}{{x}^{2}-5x - 24}[\/latex]<\/p>\n

8.\u00a0[latex]\\frac{-x - 24}{{x}^{2}-2x - 24}[\/latex]<\/p>\n

9. [latex]\\frac{10x+47}{{x}^{2}+7x+10}[\/latex]<\/p>\n

10.\u00a0[latex]\\frac{x}{6{x}^{2}+25x+25}[\/latex]<\/p>\n

11. [latex]\\frac{32x - 11}{20{x}^{2}-13x+2}[\/latex]<\/p>\n

12.\u00a0[latex]\\frac{x+1}{{x}^{2}+7x+10}[\/latex]<\/p>\n

13. [latex]\\frac{5x}{{x}^{2}-9}[\/latex]<\/p>\n

14.\u00a0[latex]\\frac{10x}{{x}^{2}-25}[\/latex]<\/p>\n

15. [latex]\\frac{6x}{{x}^{2}-4}[\/latex]<\/p>\n

16.\u00a0[latex]\\frac{2x - 3}{{x}^{2}-6x+5}[\/latex]<\/p>\n

17. [latex]\\frac{4x - 1}{{x}^{2}-x - 6}[\/latex]<\/p>\n

18.\u00a0[latex]\\frac{4x+3}{{x}^{2}+8x+15}[\/latex]<\/p>\n

19. [latex]\\frac{3x - 1}{{x}^{2}-5x+6}[\/latex]<\/p>\n

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.<\/p>\n

20. [latex]\\frac{-5x - 19}{{\\left(x+4\\right)}^{2}}[\/latex]<\/p>\n

21. [latex]\\frac{x}{{\\left(x - 2\\right)}^{2}}[\/latex]<\/p>\n

22.\u00a0[latex]\\frac{7x+14}{{\\left(x+3\\right)}^{2}}[\/latex]<\/p>\n

23. [latex]\\frac{-24x - 27}{{\\left(4x+5\\right)}^{2}}[\/latex]<\/p>\n

24.\u00a0[latex]\\frac{-24x - 27}{{\\left(6x - 7\\right)}^{2}}[\/latex]<\/p>\n

25. [latex]\\frac{5-x}{{\\left(x - 7\\right)}^{2}}[\/latex]<\/p>\n

26.\u00a0[latex]\\frac{5x+14}{2{x}^{2}+12x+18}[\/latex]<\/p>\n

27. [latex]\\frac{5{x}^{2}+20x+8}{2x{\\left(x+1\\right)}^{2}}[\/latex]<\/p>\n

28.\u00a0[latex]\\frac{4{x}^{2}+55x+25}{5x{\\left(3x+5\\right)}^{2}}[\/latex]<\/p>\n

29. [latex]\\frac{54{x}^{3}+127{x}^{2}+80x+16}{2{x}^{2}{\\left(3x+2\\right)}^{2}}[\/latex]<\/p>\n

30.\u00a0[latex]\\frac{{x}^{3}-5{x}^{2}+12x+144}{{x}^{2}\\left({x}^{2}+12x+36\\right)}[\/latex]<\/p>\n

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.<\/p>\n

31. [latex]\\frac{4{x}^{2}+6x+11}{\\left(x+2\\right)\\left({x}^{2}+x+3\\right)}[\/latex]<\/p>\n

32.\u00a0[latex]\\frac{4{x}^{2}+9x+23}{\\left(x - 1\\right)\\left({x}^{2}+6x+11\\right)}[\/latex]<\/p>\n

33. [latex]\\frac{-2{x}^{2}+10x+4}{\\left(x - 1\\right)\\left({x}^{2}+3x+8\\right)}[\/latex]<\/p>\n

34.\u00a0[latex]\\frac{{x}^{2}+3x+1}{\\left(x+1\\right)\\left({x}^{2}+5x - 2\\right)}[\/latex]<\/p>\n

35. [latex]\\frac{4{x}^{2}+17x - 1}{\\left(x+3\\right)\\left({x}^{2}+6x+1\\right)}[\/latex]<\/p>\n

36.\u00a0[latex]\\frac{4{x}^{2}}{\\left(x+5\\right)\\left({x}^{2}+7x - 5\\right)}[\/latex]<\/p>\n

37. [latex]\\frac{4{x}^{2}+5x+3}{{x}^{3}-1}[\/latex]<\/p>\n

38.\u00a0[latex]\\frac{-5{x}^{2}+18x - 4}{{x}^{3}+8}[\/latex]<\/p>\n

39. [latex]\\frac{3{x}^{2}-7x+33}{{x}^{3}+27}[\/latex]<\/p>\n

40.\u00a0[latex]\\frac{{x}^{2}+2x+40}{{x}^{3}-125}[\/latex]<\/p>\n

41. [latex]\\frac{4{x}^{2}+4x+12}{8{x}^{3}-27}[\/latex]<\/p>\n

42.\u00a0[latex]\\frac{-50{x}^{2}+5x - 3}{125{x}^{3}-1}[\/latex]<\/p>\n

43. [latex]\\frac{-2{x}^{3}-30{x}^{2}+36x+216}{{x}^{4}+216x}[\/latex]<\/p>\n

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.<\/p>\n

44. [latex]\\frac{3{x}^{3}+2{x}^{2}+14x+15}{{\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/p>\n

45. [latex]\\frac{{x}^{3}+6{x}^{2}+5x+9}{{\\left({x}^{2}+1\\right)}^{2}}[\/latex]<\/p>\n

46.\u00a0[latex]\\frac{{x}^{3}-{x}^{2}+x - 1}{{\\left({x}^{2}-3\\right)}^{2}}[\/latex]<\/p>\n

47. [latex]\\frac{{x}^{2}+5x+5}{{\\left(x+2\\right)}^{2}}[\/latex]<\/p>\n

48.\u00a0[latex]\\frac{{x}^{3}+2{x}^{2}+4x}{{\\left({x}^{2}+2x+9\\right)}^{2}}[\/latex]<\/p>\n

49. [latex]\\frac{{x}^{2}+25}{{\\left({x}^{2}+3x+25\\right)}^{2}}[\/latex]<\/p>\n

50.\u00a0[latex]\\frac{2{x}^{3}+11x+7x+70}{{\\left(2{x}^{2}+x+14\\right)}^{2}}[\/latex]<\/p>\n

51. [latex]\\frac{5x+2}{x{\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/p>\n

52.\u00a0[latex]\\frac{{x}^{4}+{x}^{3}+8{x}^{2}+6x+36}{x{\\left({x}^{2}+6\\right)}^{2}}[\/latex]<\/p>\n

53. [latex]\\frac{2x - 9}{{\\left({x}^{2}-x\\right)}^{2}}[\/latex]<\/p>\n

54.\u00a0[latex]\\frac{5{x}^{3}-2x+1}{{\\left({x}^{2}+2x\\right)}^{2}}[\/latex]<\/p>\n

For the following exercises, find the partial fraction expansion.<\/p>\n

55. [latex]\\frac{{x}^{2}+4}{{\\left(x+1\\right)}^{3}}[\/latex]<\/p>\n

56.\u00a0[latex]\\frac{{x}^{3}-4{x}^{2}+5x+4}{{\\left(x - 2\\right)}^{3}}[\/latex]<\/p>\n

For the following exercises, perform the operation and then find the partial fraction decomposition.<\/p>\n

57. [latex]\\frac{7}{x+8}+\\frac{5}{x - 2}-\\frac{x - 1}{{x}^{2}-6x - 16}[\/latex]<\/p>\n

58.\u00a0[latex]\\frac{1}{x - 4}-\\frac{3}{x+6}-\\frac{2x+7}{{x}^{2}+2x - 24}[\/latex]<\/p>\n

59. [latex]\\frac{2x}{{x}^{2}-16}-\\frac{1 - 2x}{{x}^{2}+6x+8}-\\frac{x - 5}{{x}^{2}-4x}[\/latex]<\/p>\n\n\t\t\t

\n\t\t\t

Candela Citations<\/h3>\n\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
CC licensed content, Shared previously<\/div>