{"id":15551,"date":"2019-09-04T20:58:02","date_gmt":"2019-09-04T20:58:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&p=15551"},"modified":"2025-02-05T05:19:22","modified_gmt":"2025-02-05T05:19:22","slug":"problem-set-16-dividing-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/problem-set-16-dividing-polynomials\/","title":{"raw":"Problem Set 16: Dividing Polynomials","rendered":"Problem Set 16: Dividing Polynomials"},"content":{"raw":"1. If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?\r\n\r\n2.\u00a0If a polynomial of degree n<\/em>\u00a0is divided by a binomial of degree 1, what is the degree of the quotient?\r\n\r\nFor the following exercises, use long division to divide. Specify the quotient and the remainder.\r\n\r\n3. [latex]\\left({x}^{2}+5x - 1\\right)\\div \\left(x - 1\\right)[\/latex]\r\n\r\n4.\u00a0[latex]\\left(2{x}^{2}-9x - 5\\right)\\div \\left(x - 5\\right)[\/latex]\r\n\r\n5. [latex]\\left(3{x}^{2}+23x+14\\right)\\div \\left(x+7\\right)[\/latex]\r\n\r\n6.\u00a0[latex]\\left(4{x}^{2}-10x+6\\right)\\div \\left(4x+2\\right)[\/latex]\r\n\r\n7. [latex]\\left(6{x}^{2}-25x - 25\\right)\\div \\left(6x+5\\right)[\/latex]\r\n\r\n8.\u00a0[latex]\\left(-{x}^{2}-1\\right)\\div \\left(x+1\\right)[\/latex]\r\n\r\n9. [latex]\\left(2{x}^{2}-3x+2\\right)\\div \\left(x+2\\right)[\/latex]\r\n\r\n10.\u00a0[latex]\\left({x}^{3}-126\\right)\\div \\left(x - 5\\right)[\/latex]\r\n\r\n11. [latex]\\left(3{x}^{2}-5x+4\\right)\\div \\left(3x+1\\right)[\/latex]\r\n\r\n12.\u00a0[latex]\\left({x}^{3}-3{x}^{2}+5x - 6\\right)\\div \\left(x - 2\\right)[\/latex]\r\n\r\n13. [latex]\\left(2{x}^{3}+3{x}^{2}-4x+15\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\nFor the following exercises, use synthetic division to find the quotient.\r\n\r\n14. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\n15. [latex]\\left(2{x}^{3}-6{x}^{2}-7x+6\\right)\\div \\left(x - 4\\right)[\/latex]\r\n\r\n16.\u00a0[latex]\\left(6{x}^{3}-10{x}^{2}-7x - 15\\right)\\div \\left(x+1\\right)[\/latex]\r\n\r\n17. [latex]\\left(4{x}^{3}-12{x}^{2}-5x - 1\\right)\\div \\left(2x+1\\right)[\/latex]\r\n\r\n18.\u00a0[latex]\\left(9{x}^{3}-9{x}^{2}+18x+5\\right)\\div \\left(3x - 1\\right)[\/latex]\r\n\r\n19. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\n20.\u00a0[latex]\\left(-6{x}^{3}+{x}^{2}-4\\right)\\div \\left(2x - 3\\right)[\/latex]\r\n\r\n21. [latex]\\left(2{x}^{3}+7{x}^{2}-13x - 3\\right)\\div \\left(2x - 3\\right)[\/latex]\r\n\r\n22.\u00a0[latex]\\left(3{x}^{3}-5{x}^{2}+2x+3\\right)\\div \\left(x+2\\right)[\/latex]\r\n\r\n23. [latex]\\left(4{x}^{3}-5{x}^{2}+13\\right)\\div \\left(x+4\\right)[\/latex]\r\n\r\n24.\u00a0[latex]\\left({x}^{3}-3x+2\\right)\\div \\left(x+2\\right)[\/latex]\r\n\r\n25. [latex]\\left({x}^{3}-21{x}^{2}+147x - 343\\right)\\div \\left(x - 7\\right)[\/latex]\r\n\r\n26.\u00a0[latex]\\left({x}^{3}-15{x}^{2}+75x - 125\\right)\\div \\left(x - 5\\right)[\/latex]\r\n\r\n27. [latex]\\left(9{x}^{3}-x+2\\right)\\div \\left(3x - 1\\right)[\/latex]\r\n\r\n28.\u00a0[latex]\\left(6{x}^{3}-{x}^{2}+5x+2\\right)\\div \\left(3x+1\\right)[\/latex]\r\n\r\n29. [latex]\\left({x}^{4}+{x}^{3}-3{x}^{2}-2x+1\\right)\\div \\left(x+1\\right)[\/latex]\r\n\r\n30.\u00a0[latex]\\left({x}^{4}-3{x}^{2}+1\\right)\\div \\left(x - 1\\right)[\/latex]\r\n\r\n31. [latex]\\left({x}^{4}+2{x}^{3}-3{x}^{2}+2x+6\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\n32.\u00a0[latex]\\left({x}^{4}-10{x}^{3}+37{x}^{2}-60x+36\\right)\\div \\left(x - 2\\right)[\/latex]\r\n\r\n33. [latex]\\left({x}^{4}-8{x}^{3}+24{x}^{2}-32x+16\\right)\\div \\left(x - 2\\right)[\/latex]\r\n\r\n34.\u00a0[latex]\\left({x}^{4}+5{x}^{3}-3{x}^{2}-13x+10\\right)\\div \\left(x+5\\right)[\/latex]\r\n\r\n35. [latex]\\left({x}^{4}-12{x}^{3}+54{x}^{2}-108x+81\\right)\\div \\left(x - 3\\right)[\/latex]\r\n\r\n36.\u00a0[latex]\\left(4{x}^{4}-2{x}^{3}-4x+2\\right)\\div \\left(2x - 1\\right)[\/latex]\r\n\r\n37. [latex]\\left(4{x}^{4}+2{x}^{3}-4{x}^{2}+2x+2\\right)\\div \\left(2x+1\\right)[\/latex]\r\n\r\nFor the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.\r\n\r\n38. Factor is [latex]{x}^{2}-x+3[\/latex]\r\n\"Graph\r\n\r\n39. Factor is [latex]\\left({x}^{2}+2x+4\\right)[\/latex]\r\n\"Graph\r\n\r\n40. Factor is [latex]{x}^{2}+2x+5[\/latex]\r\n\"Graph\r\n\r\n41. Factor is [latex]{x}^{2}+x+1[\/latex]\r\n\"Graph\r\n\r\n42.\u00a0Factor is [latex]{x}^{2}+2x+2[\/latex]\r\n\"Graph\r\n\r\nFor the following exercises, use synthetic division to find the quotient and remainder.\r\n\r\n43. [latex]\\frac{4{x}^{3}-33}{x - 2}[\/latex]\r\n\r\n44.\u00a0[latex]\\frac{2{x}^{3}+25}{x+3}[\/latex]\r\n\r\n45. [latex]\\frac{3{x}^{3}+2x - 5}{x - 1}[\/latex]\r\n\r\n46.\u00a0[latex]\\frac{-4{x}^{3}-{x}^{2}-12}{x+4}[\/latex]\r\n\r\n47. [latex]\\frac{{x}^{4}-22}{x+2}[\/latex]\r\n\r\nFor the following exercises, use a calculator with CAS to answer the questions.\r\n\r\n48. Consider [latex]\\frac{{x}^{k}-1}{x - 1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if k<\/em> = 4?\r\n\r\n49. Consider [latex]\\frac{{x}^{k}+1}{x+1}[\/latex] for [latex]k=1, 3, 5[\/latex]. What do you expect the result to be if\u00a0k<\/em> = 7?\r\n\r\n50.\u00a0Consider [latex]\\frac{{x}^{4}-{k}^{4}}{x-k}[\/latex] for [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0k<\/em> = 4?\r\n\r\n51. Consider [latex]\\frac{{x}^{k}}{x+1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0k<\/em> = 4?\r\n\r\n52.\u00a0Consider [latex]\\frac{{x}^{k}}{x - 1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0k<\/em> = 4?\r\n\r\nFor the following exercises, use synthetic division to determine the quotient involving a complex number.\r\n\r\n53. [latex]\\frac{x+1}{x-i}[\/latex]\r\n\r\n54.\u00a0[latex]\\frac{{x}^{2}+1}{x-i}[\/latex]\r\n\r\n55. [latex]\\frac{x+1}{x+i}[\/latex]\r\n\r\n56.\u00a0[latex]\\frac{{x}^{2}+1}{x+i}[\/latex]\r\n\r\n57. [latex]\\frac{{x}^{3}+1}{x-i}[\/latex]\r\n\r\nFor the following exercises, use the given length and area of a rectangle to express the width algebraically.\r\n\r\n58. Length is [latex]x+5[\/latex], area is [latex]2{x}^{2}+9x - 5[\/latex].\r\n\r\n59. Length is [latex]2x\\text{ }+\\text{ }5[\/latex], area is [latex]4{x}^{3}+10{x}^{2}+6x+15[\/latex]\r\n\r\n60.\u00a0Length is [latex]3x - 4[\/latex], area is [latex]6{x}^{4}-8{x}^{3}+9{x}^{2}-9x - 4[\/latex]\r\n\r\nFor the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.\r\n\r\n61. Volume is [latex]12{x}^{3}+20{x}^{2}-21x - 36[\/latex], length is [latex]2x+3[\/latex], width is [latex]3x - 4[\/latex].\r\n\r\n62.\u00a0Volume is [latex]18{x}^{3}-21{x}^{2}-40x+48[\/latex], length is [latex]3x - 4[\/latex],\u00a0width is [latex]3x - 4[\/latex].\r\n\r\n63. Volume is [latex]10{x}^{3}+27{x}^{2}+2x - 24[\/latex], length is [latex]5x - 4[\/latex],\u00a0width is [latex]2x+3[\/latex].\r\n\r\n64.\u00a0Volume is [latex]10{x}^{3}+30{x}^{2}-8x - 24[\/latex], length is 2, width is [latex]x+3[\/latex].\r\n\r\nFor the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.\r\n\r\n65. Volume is [latex]\\pi \\left(25{x}^{3}-65{x}^{2}-29x - 3\\right)[\/latex], radius is [latex]5x+1[\/latex].\r\n\r\n66.\u00a0Volume is [latex]\\pi \\left(4{x}^{3}+12{x}^{2}-15x - 50\\right)[\/latex], radius is [latex]2x+5[\/latex].\r\n\r\n67. Volume is [latex]\\pi \\left(3{x}^{4}+24{x}^{3}+46{x}^{2}-16x - 32\\right)[\/latex], radius is [latex]x+4[\/latex].","rendered":"

1. If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?<\/p>\n

2.\u00a0If a polynomial of degree n<\/em>\u00a0is divided by a binomial of degree 1, what is the degree of the quotient?<\/p>\n

For the following exercises, use long division to divide. Specify the quotient and the remainder.<\/p>\n

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

4.\u00a0[latex]\\left(2{x}^{2}-9x - 5\\right)\\div \\left(x - 5\\right)[\/latex]<\/p>\n

5. [latex]\\left(3{x}^{2}+23x+14\\right)\\div \\left(x+7\\right)[\/latex]<\/p>\n

6.\u00a0[latex]\\left(4{x}^{2}-10x+6\\right)\\div \\left(4x+2\\right)[\/latex]<\/p>\n

7. [latex]\\left(6{x}^{2}-25x - 25\\right)\\div \\left(6x+5\\right)[\/latex]<\/p>\n

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

9. [latex]\\left(2{x}^{2}-3x+2\\right)\\div \\left(x+2\\right)[\/latex]<\/p>\n

10.\u00a0[latex]\\left({x}^{3}-126\\right)\\div \\left(x - 5\\right)[\/latex]<\/p>\n

11. [latex]\\left(3{x}^{2}-5x+4\\right)\\div \\left(3x+1\\right)[\/latex]<\/p>\n

12.\u00a0[latex]\\left({x}^{3}-3{x}^{2}+5x - 6\\right)\\div \\left(x - 2\\right)[\/latex]<\/p>\n

13. [latex]\\left(2{x}^{3}+3{x}^{2}-4x+15\\right)\\div \\left(x+3\\right)[\/latex]<\/p>\n

For the following exercises, use synthetic division to find the quotient.<\/p>\n

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

15. [latex]\\left(2{x}^{3}-6{x}^{2}-7x+6\\right)\\div \\left(x - 4\\right)[\/latex]<\/p>\n

16.\u00a0[latex]\\left(6{x}^{3}-10{x}^{2}-7x - 15\\right)\\div \\left(x+1\\right)[\/latex]<\/p>\n

17. [latex]\\left(4{x}^{3}-12{x}^{2}-5x - 1\\right)\\div \\left(2x+1\\right)[\/latex]<\/p>\n

18.\u00a0[latex]\\left(9{x}^{3}-9{x}^{2}+18x+5\\right)\\div \\left(3x - 1\\right)[\/latex]<\/p>\n

19. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]<\/p>\n

20.\u00a0[latex]\\left(-6{x}^{3}+{x}^{2}-4\\right)\\div \\left(2x - 3\\right)[\/latex]<\/p>\n

21. [latex]\\left(2{x}^{3}+7{x}^{2}-13x - 3\\right)\\div \\left(2x - 3\\right)[\/latex]<\/p>\n

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

23. [latex]\\left(4{x}^{3}-5{x}^{2}+13\\right)\\div \\left(x+4\\right)[\/latex]<\/p>\n

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

25. [latex]\\left({x}^{3}-21{x}^{2}+147x - 343\\right)\\div \\left(x - 7\\right)[\/latex]<\/p>\n

26.\u00a0[latex]\\left({x}^{3}-15{x}^{2}+75x - 125\\right)\\div \\left(x - 5\\right)[\/latex]<\/p>\n

27. [latex]\\left(9{x}^{3}-x+2\\right)\\div \\left(3x - 1\\right)[\/latex]<\/p>\n

28.\u00a0[latex]\\left(6{x}^{3}-{x}^{2}+5x+2\\right)\\div \\left(3x+1\\right)[\/latex]<\/p>\n

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

30.\u00a0[latex]\\left({x}^{4}-3{x}^{2}+1\\right)\\div \\left(x - 1\\right)[\/latex]<\/p>\n

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

32.\u00a0[latex]\\left({x}^{4}-10{x}^{3}+37{x}^{2}-60x+36\\right)\\div \\left(x - 2\\right)[\/latex]<\/p>\n

33. [latex]\\left({x}^{4}-8{x}^{3}+24{x}^{2}-32x+16\\right)\\div \\left(x - 2\\right)[\/latex]<\/p>\n

34.\u00a0[latex]\\left({x}^{4}+5{x}^{3}-3{x}^{2}-13x+10\\right)\\div \\left(x+5\\right)[\/latex]<\/p>\n

35. [latex]\\left({x}^{4}-12{x}^{3}+54{x}^{2}-108x+81\\right)\\div \\left(x - 3\\right)[\/latex]<\/p>\n

36.\u00a0[latex]\\left(4{x}^{4}-2{x}^{3}-4x+2\\right)\\div \\left(2x - 1\\right)[\/latex]<\/p>\n

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

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.<\/p>\n

38. Factor is [latex]{x}^{2}-x+3[\/latex]
\n\"Graph<\/p>\n

39. Factor is [latex]\\left({x}^{2}+2x+4\\right)[\/latex]
\n\"Graph<\/p>\n

40. Factor is [latex]{x}^{2}+2x+5[\/latex]
\n\"Graph<\/p>\n

41. Factor is [latex]{x}^{2}+x+1[\/latex]
\n\"Graph<\/p>\n

42.\u00a0Factor is [latex]{x}^{2}+2x+2[\/latex]
\n\"Graph<\/p>\n

For the following exercises, use synthetic division to find the quotient and remainder.<\/p>\n

43. [latex]\\frac{4{x}^{3}-33}{x - 2}[\/latex]<\/p>\n

44.\u00a0[latex]\\frac{2{x}^{3}+25}{x+3}[\/latex]<\/p>\n

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

46.\u00a0[latex]\\frac{-4{x}^{3}-{x}^{2}-12}{x+4}[\/latex]<\/p>\n

47. [latex]\\frac{{x}^{4}-22}{x+2}[\/latex]<\/p>\n

For the following exercises, use a calculator with CAS to answer the questions.<\/p>\n

48. Consider [latex]\\frac{{x}^{k}-1}{x - 1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if k<\/em> = 4?<\/p>\n

49. Consider [latex]\\frac{{x}^{k}+1}{x+1}[\/latex] for [latex]k=1, 3, 5[\/latex]. What do you expect the result to be if\u00a0k<\/em> = 7?<\/p>\n

50.\u00a0Consider [latex]\\frac{{x}^{4}-{k}^{4}}{x-k}[\/latex] for [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0k<\/em> = 4?<\/p>\n

51. Consider [latex]\\frac{{x}^{k}}{x+1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0k<\/em> = 4?<\/p>\n

52.\u00a0Consider [latex]\\frac{{x}^{k}}{x - 1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0k<\/em> = 4?<\/p>\n

For the following exercises, use synthetic division to determine the quotient involving a complex number.<\/p>\n

53. [latex]\\frac{x+1}{x-i}[\/latex]<\/p>\n

54.\u00a0[latex]\\frac{{x}^{2}+1}{x-i}[\/latex]<\/p>\n

55. [latex]\\frac{x+1}{x+i}[\/latex]<\/p>\n

56.\u00a0[latex]\\frac{{x}^{2}+1}{x+i}[\/latex]<\/p>\n

57. [latex]\\frac{{x}^{3}+1}{x-i}[\/latex]<\/p>\n

For the following exercises, use the given length and area of a rectangle to express the width algebraically.<\/p>\n

58. Length is [latex]x+5[\/latex], area is [latex]2{x}^{2}+9x - 5[\/latex].<\/p>\n

59. Length is [latex]2x\\text{ }+\\text{ }5[\/latex], area is [latex]4{x}^{3}+10{x}^{2}+6x+15[\/latex]<\/p>\n

60.\u00a0Length is [latex]3x - 4[\/latex], area is [latex]6{x}^{4}-8{x}^{3}+9{x}^{2}-9x - 4[\/latex]<\/p>\n

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.<\/p>\n

61. Volume is [latex]12{x}^{3}+20{x}^{2}-21x - 36[\/latex], length is [latex]2x+3[\/latex], width is [latex]3x - 4[\/latex].<\/p>\n

62.\u00a0Volume is [latex]18{x}^{3}-21{x}^{2}-40x+48[\/latex], length is [latex]3x - 4[\/latex],\u00a0width is [latex]3x - 4[\/latex].<\/p>\n

63. Volume is [latex]10{x}^{3}+27{x}^{2}+2x - 24[\/latex], length is [latex]5x - 4[\/latex],\u00a0width is [latex]2x+3[\/latex].<\/p>\n

64.\u00a0Volume is [latex]10{x}^{3}+30{x}^{2}-8x - 24[\/latex], length is 2, width is [latex]x+3[\/latex].<\/p>\n

For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.<\/p>\n

65. Volume is [latex]\\pi \\left(25{x}^{3}-65{x}^{2}-29x - 3\\right)[\/latex], radius is [latex]5x+1[\/latex].<\/p>\n

66.\u00a0Volume is [latex]\\pi \\left(4{x}^{3}+12{x}^{2}-15x - 50\\right)[\/latex], radius is [latex]2x+5[\/latex].<\/p>\n

67. Volume is [latex]\\pi \\left(3{x}^{4}+24{x}^{3}+46{x}^{2}-16x - 32\\right)[\/latex], radius is [latex]x+4[\/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>