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. $\left({x}^{2}+5x - 1\right)\div \left(x - 1\right)$

4. $\left(2{x}^{2}-9x - 5\right)\div \left(x - 5\right)$

5. $\left(3{x}^{2}+23x+14\right)\div \left(x+7\right)$

6. $\left(4{x}^{2}-10x+6\right)\div \left(4x+2\right)$

7. $\left(6{x}^{2}-25x - 25\right)\div \left(6x+5\right)$

8. $\left(-{x}^{2}-1\right)\div \left(x+1\right)$

9. $\left(2{x}^{2}-3x+2\right)\div \left(x+2\right)$

10. $\left({x}^{3}-126\right)\div \left(x - 5\right)$

11. $\left(3{x}^{2}-5x+4\right)\div \left(3x+1\right)$

12. $\left({x}^{3}-3{x}^{2}+5x - 6\right)\div \left(x - 2\right)$

13. $\left(2{x}^{3}+3{x}^{2}-4x+15\right)\div \left(x+3\right)$

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

14. $\left(3{x}^{3}-2{x}^{2}+x - 4\right)\div \left(x+3\right)$

15. $\left(2{x}^{3}-6{x}^{2}-7x+6\right)\div \left(x - 4\right)$

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

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

18. $\left(9{x}^{3}-9{x}^{2}+18x+5\right)\div \left(3x - 1\right)$

19. $\left(3{x}^{3}-2{x}^{2}+x - 4\right)\div \left(x+3\right)$

20. $\left(-6{x}^{3}+{x}^{2}-4\right)\div \left(2x - 3\right)$

21. $\left(2{x}^{3}+7{x}^{2}-13x - 3\right)\div \left(2x - 3\right)$

22. $\left(3{x}^{3}-5{x}^{2}+2x+3\right)\div \left(x+2\right)$

23. $\left(4{x}^{3}-5{x}^{2}+13\right)\div \left(x+4\right)$

24. $\left({x}^{3}-3x+2\right)\div \left(x+2\right)$

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

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

27. $\left(9{x}^{3}-x+2\right)\div \left(3x - 1\right)$

28. $\left(6{x}^{3}-{x}^{2}+5x+2\right)\div \left(3x+1\right)$

29. $\left({x}^{4}+{x}^{3}-3{x}^{2}-2x+1\right)\div \left(x+1\right)$

30. $\left({x}^{4}-3{x}^{2}+1\right)\div \left(x - 1\right)$

31. $\left({x}^{4}+2{x}^{3}-3{x}^{2}+2x+6\right)\div \left(x+3\right)$

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

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

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

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

36. $\left(4{x}^{4}-2{x}^{3}-4x+2\right)\div \left(2x - 1\right)$

37. $\left(4{x}^{4}+2{x}^{3}-4{x}^{2}+2x+2\right)\div \left(2x+1\right)$

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 ${x}^{2}-x+3$

39. Factor is $\left({x}^{2}+2x+4\right)$

40. Factor is ${x}^{2}+2x+5$

41. Factor is ${x}^{2}+x+1$

42. Factor is ${x}^{2}+2x+2$

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

43. $\frac{4{x}^{3}-33}{x - 2}$

44. $\frac{2{x}^{3}+25}{x+3}$

45. $\frac{3{x}^{3}+2x - 5}{x - 1}$

46. $\frac{-4{x}^{3}-{x}^{2}-12}{x+4}$

47. $\frac{{x}^{4}-22}{x+2}$

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

48. Consider $\frac{{x}^{k}-1}{x - 1}$ with $k=1, 2, 3$. What do you expect the result to be if k = 4?

49. Consider $\frac{{x}^{k}+1}{x+1}$ for $k=1, 3, 5$. What do you expect the result to be if k = 7?

50. Consider $\frac{{x}^{4}-{k}^{4}}{x-k}$ for $k=1, 2, 3$. What do you expect the result to be if k = 4?

51. Consider $\frac{{x}^{k}}{x+1}$ with $k=1, 2, 3$. What do you expect the result to be if k = 4?

52. Consider $\frac{{x}^{k}}{x - 1}$ with $k=1, 2, 3$. 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. $\frac{x+1}{x-i}$

54. $\frac{{x}^{2}+1}{x-i}$

55. $\frac{x+1}{x+i}$

56. $\frac{{x}^{2}+1}{x+i}$

57. $\frac{{x}^{3}+1}{x-i}$

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

58. Length is $x+5$, area is $2{x}^{2}+9x - 5$.

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

60. Length is $3x - 4$, area is $6{x}^{4}-8{x}^{3}+9{x}^{2}-9x - 4$

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 $12{x}^{3}+20{x}^{2}-21x - 36$, length is $2x+3$, width is $3x - 4$.

62. Volume is $18{x}^{3}-21{x}^{2}-40x+48$, length is $3x - 4$, width is $3x - 4$.

63. Volume is $10{x}^{3}+27{x}^{2}+2x - 24$, length is $5x - 4$, width is $2x+3$.

64. Volume is $10{x}^{3}+30{x}^{2}-8x - 24$, length is 2, width is $x+3$.

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

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

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

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