1. If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?

2. If a polynomial of degree n is divided by a binomial of degree 1, what is the degree of the quotient?

For the following exercises, use long division to divide. Specify the quotient and the remainder.

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

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

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

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

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

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

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

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

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

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

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

For the following exercises, use synthetic division to find the quotient.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

38. Factor is [latex]{x}^{2}-x+3\\[/latex]

39. Factor is [latex]\left({x}^{2}+2x+4\right)\\[/latex]

40. Factor is [latex]{x}^{2}+2x+5\\[/latex]

41. Factor is [latex]{x}^{2}+x+1\\[/latex]

42. Factor is [latex]{x}^{2}+2x+2\\[/latex]

For the following exercises, use synthetic division to find the quotient and remainder.

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

44. [latex]\frac{2{x}^{3}+25}{x+3}\\[/latex]

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

46. [latex]\frac{-4{x}^{3}-{x}^{2}-12}{x+4}\\[/latex]

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

For the following exercises, use a calculator with CAS to answer the questions.

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 = 4?

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 k = 7?

50. Consider [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 k = 4?

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 k = 4?

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

For the following exercises, use synthetic division to determine the quotient involving a complex number.

53. [latex]\frac{x+1}{x-i}\\[/latex]

54. [latex]\frac{{x}^{2}+1}{x-i}\\[/latex]

55. [latex]\frac{x+1}{x+i}\\[/latex]

56. [latex]\frac{{x}^{2}+1}{x+i}\\[/latex]

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

For the following exercises, use the given length and area of a rectangle to express the width algebraically.

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

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

60. Length is [latex]3x - 4\\[/latex], area is [latex]6{x}^{4}-8{x}^{3}+9{x}^{2}-9x - 4\\[/latex]

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.

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].

62. Volume is [latex]18{x}^{3}-21{x}^{2}-40x+48\\[/latex], length is [latex]3x - 4\\[/latex], width is [latex]3x - 4\\[/latex].

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

64. Volume is [latex]10{x}^{3}+30{x}^{2}-8x - 24\\[/latex], length is 2, width is [latex]x+3\\[/latex].

For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.

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

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

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].