3.5e: Exercises - Division of Polynomials

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A: Concepts

Exercise $\PageIndex{A}$

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?

Answers to odd exercises:: 1. The binomial is a factor of the polynomial.

B: Perform Polynomial Long Division

Exercise $\PageIndex{B}$

$ \bigstar $ Use long division to divide. Also specify the quotient and the remainder.

3) $(x^2+5x−1)÷(x−1)$

4) $(2x^2−9x−5)÷(x−5)$

5) $(3x^2+23x+14)÷(x+7)$

6) $(4x^2−10x+6)÷(4x+2)$

7) $(6x^2−25x−25)÷(6x+5)$

8) $(−x^2−1)÷(x+1)$

9) $(2x^2−3x+2)÷(x+2)$

10) $(x^3−126)÷(x−5)$

11) $(3x^2−5x+4)÷(3x+1)$

12) $(x^3−3x^2+5x−6)÷(x−2)$

13) $(2x^3+3x^2−4x+15)÷(x+3)$

$ \bigstar $ Divide.

14) $\dfrac{x^{3}-5 x^{2}+x+15}{x-3}$

15) $\dfrac{y^{3}-4 y^{2}+6 y-4}{y-2}$

16) $\dfrac{x^{5}+3 x+2}{x^{3}+2 x+1}$

17) $\dfrac{2 z^{3}+5 z+8}{z+1}$

18) $\dfrac{3 x^{5}-4 x^{3}+3 x^{2}+12 x-10}{x^{2}+2 x-1}$

19) $\dfrac{2 y^{5}-3 y^{4}-y^{2}+y+4}{y^{2}+1}$

20) $\dfrac{3 y^{3}-4 y^{2}-3}{y^{2}+5 y+2}$

21) $\dfrac{5 x^{4}-3 x^{2}+2}{x^{2}-3 x+5}$

Answers to odd exercises:

3. $\mathrm{x+6+\dfrac{5}{x−1},
quotient: x+6, remainder: 5}$

5. $\mathrm{3x+2,
quotient: 3x+2, remainder: 0}$

7. $\mathrm{x−5,
quotient: x−5, remainder: 0}$

9. $\mathrm{2x−7+\dfrac{16}{x+2},
quotient: 2x−7, remainder: 16}$

11. $\mathrm{x−2+\dfrac{6}{3x+1},
quotient: x−2, remainder: 6}$

13. $\mathrm{2x^2−3x+5,
quotient: 2x^2−3x+5, remainder: 0}$

15. $ y^2-2y+2 $

17. $ 2z^2-2z+7 + \dfrac{1}{z+1} $

19. $ 2y^3 -3y^2 -2y +2 +\dfrac{3y+2}{y^2+1} $

21. $ 5x^2 + 15x + 17 + \dfrac{-24x-83}{x^2-3 x+5} $

C: Use Long Division to Rewrite a Polynomial

Exercise $\PageIndex{C}$

$ \bigstar $ Use polynomial long division to perform the indicated division. Write the polynomial dividend in the form $p(x) = d(x)q(x) + r(x)$.

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

24) $\dfrac{x^{3}-4 x^{2}-3 x-10}{x^{2}+x+2}$

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

26) $\dfrac{2 x^{3}-3 x^{2}+7 x-3}{x^{2}-x+3}$

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

28) $\dfrac{x^{4}+2 x^{3}-x^{2}+x+6}{x+2}$

29. $\left(-x^{5} + 7x^{3} - x \right) \div \left(x^{3} - x^{2} + 1 \right)$

30) $\dfrac{x^{4}+x^{3}+5 x^{2}+3 x+6}{x^{2}+x-1}$

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

32) $\dfrac{x^{4}+2 x^{3}+4 x^{2}+3 x+2}{x^{2}+x+2}$

33. $\left(4x^2 - x - 23 \right) \div \left(x^{2} - 1 \right)$

34) $\dfrac{2 x^{4}+3 x^{3}+3 x^{2}-5 x-3}{2 x^{2}-x-1}$

Answers to odd exercises:

23. $4x^2+3x-1 = (x-3)(4x+15) + 44$

25. $2x^3-x+1 = \left(x^2+x+1\right)(2x-2)+(-x+3)$

27. $5x^{4} - 3x^{3} + 2x^{2} - 1 = \left(x^{2} + 4 \right) \left(5x^{2} - 3x - 18 \right) + (12x + 71)$

29. $-x^{5} + 7x^{3} - x = \left(x^{3} - x^{2} + 1 \right) \left(-x^{2} - x + 6 \right) + \left(7x^{2} - 6 \right)$

31. $9x^{3} + 5 =(2x - 3) \left(\frac{9}{2}x^{2} + \frac{27}{4}x + \frac{81}{8} \right) + \frac{283}{8}$

33. $4x^2 - x - 23 = \left(x^{2} - 1 \right)(4) + (-x - 19)$

D: Perform Synthetic Division

Exercise $\PageIndex{D}$

$ \bigstar $ Use synthetic division to divide. Also state the quotient and remainder.

35) $\dfrac{4x^3−33}{x−2}$

36) $\dfrac{2x^3+25}{x+3}$

37) $\dfrac{3x^3+2x−5}{x−1}$

38) $\dfrac{−4x^3−x^2−12}{x+4}$

39) $\dfrac{x^4−22}{x+2}$

$ \bigstar $For the exercises below, use synthetic division to find the quotient.

40) $(3x^3−2x^2+x−4)÷(x+3)$

41) $(2x^3−6x^2−7x+6)÷(x−4)$

42) $(6x^3−10x^2−7x−15)÷(x+1)$

43) $(4x^3−12x^2−5x−1)÷(2x+1)$

44) $(9x^3−9x^2+18x+5)÷(3x−1)$

45) $(3x^3−2x^2+x−4)÷(x+3)$

46) $(−6x^3+x^2−4)÷(2x−3)$

47) $(2x^3+7x^2−13x−3)÷(2x−3)$

48) $(3x^3−5x^2+2x+3)÷(x+2)$

49) $(4x^3−5x^2+13)÷(x+4)$

50) $(x^3−3x+2)÷(x+2)$

51) $(x^3−21x^2+147x−343)÷(x−7)$

52) $(x^3−15x^2+75x−125)÷(x−5)$

53) $(9x^3−x+2)÷(3x−1)$

54) $(6x^3−x^2+5x+2)÷(3x+1)$

55) $(x^4+x^3−3x^2−2x+1)÷(x+1)$

56) $(x^4−3x^2+1)÷(x−1)$

57) $(x^4+2x^3−3x^2+2x+6)÷(x+3)$

58) $(x^4−10x^3+37x^2−60x+36)÷(x−2)$

59) $(x^4−8x^3+24x^2−32x+16)÷(x−2)$

60) $(x^4+5x^3−3x^2−13x+10)÷(x+5)$

61) $(x^4−12x^3+54x^2−108x+81)÷(x−3)$

62) $(4x^4−2x^3−4x+2)÷(2x−1)$

63) $(4x^4+2x^3−4x^2+2x+2)÷(2x+1)$

Answers to odd exercises:

35. $ 4x^2+8x+16 + \frac{-1}{x−2}$,
$\mathrm{Quotient: 4x^2+8x+16, remainder: −1}$

37. $ 3x^2+3x+5$ ,
$\mathrm{Quotient: 3x^2+3x+5, remainder: 0}$

39. $ x^3−2x^2+4x−8 + \frac{-6}{x+2}$,
$\mathrm{Quotient: x^3−2x^2+4x−8, remainder: −6}$

41. $2x^2+2x+1+\dfrac{10}{x−4}$

43. $2x^2−7x+1−\dfrac{2}{2x+1}$

45. $3x^2−11x+34−\dfrac{106}{x+3}$

47. $x^2+5x+1$

49. $4x^2−21x+84−\dfrac{323}{x+4}$

51. $x^2−14x+49$

53. $3x^2+x+\dfrac{2}{3x−1}$

55. $x^3−3x+1$

57. $x^3−x^2+2$

59. $x^3−6x^2+12x−8$

61. $x^3−9x^2+27x−27$

63. $2x^3−2x+2$

E: Use Synthetic Division to Rewrite a Polynomial

Exercise $\PageIndex{E}$

$ \bigstar $ For the exercises below, use synthetic division to determine whether the first expression is a factor of the second. If it is, write the second expression as a product of two factors.

64) $x−2, \; 4x^3−3x^2−8x+4$

65) $x−2, \; 3x^4−6x^3−5x+10$

66) $x+3, \; −4x^3+5x^2+8$

67) $x−2, \; 4x^4−15x^2−4$

68) $x−\dfrac{1}{2}, \; 2x^4−x^3+2x−1$

69) $x+\dfrac{1}{3}, \; 3x^4+x^3−3x+1$

$ \bigstar $ In the exercises below, use synthetic division to perform the indicated division. Write the polynomial in the form$p(x) = d(x)q(x) + r(x)$.

70. $\left(3x^2-2x+1 \right) \div \left(x-1\right)$

71. $\left(x^2-5 \right) \div \left(x-5\right)$

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

73. $\left(4x^2-5x +3\right) \div \left(x+3\right)$

74. $\left(x^3 + 8 \right) \div \left(x+2\right)$

75. $\left(4x^3 +2x-3 \right) \div \left(x -3\right)$

76. $\left(18x^2-15x-25\right) \div \left(x - \frac{5}{3} \right)$

77. $\left(4x^2-1 \right) \div \left(x - \frac{1}{2} \right)$

78. $\left(2x^3+x^2+2x+1 \right) \div \left(x + \frac{1}{2} \right)$

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

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

81. $\left(4x^4-12x^3+13x^2 -12x+9\right) \div \left(x - \frac{3}{2} \right)$

82. $\left(x^4-6x^2+9 \right) \div \left(x -\sqrt{3} \right)$

83. $\left(x^6-6x^4+12x^2-8\right) \div \left(x +\sqrt{2} \right)$

Answers to odd exercises:

65. Yes $(x−2)(3x^3−5)$

67. Yes $(x−2)(4x^3+8x^2+x+2)$

69. No

71. $\left(x^2-5 \right)= \left(x-5\right)(x+5) + 20$

73. $\left(4x^2-5x +3\right) = \left(x+3\right)(4x-17)+54$

75. $\left(4x^3 +2x-3 \right) = \left(x -3\right) \left(4x^2+12x+38\right) + 111$

77. $\left(4x^2-1 \right) = \left(x - \frac{1}{2} \right)(4x+2)+0$

79. $\left(3x^3 - x + 4 \right) = \left(x - \frac{2}{3} \right) \left(3x^2+2x+\frac{1}{3}\right) + \frac{38}{9}$

81. $\left(4x^4-12x^3+13x^2 -12x+9\right) = \left(x - \frac{3}{2} \right) \left(4x^3-6x^2+4x-6 \right)+0$

83. $\left(x^6-6x^4+12x^2-8\right) = \left(x +\sqrt{2} \right) \left(x^5-\sqrt{2} \, x^4-4x^3+4\sqrt{2} \, x^2+4x-4\sqrt{2}\right) + 0$

F: Synthetic Division with Complex Numbers

Exercise $\PageIndex{F}$

$ \bigstar $ For the exercises below, use synthetic division to determine the quotient involving a complex number.

85) $\dfrac{x+1}{x−i}$

86) $\dfrac{x^2+1}{x−i}$

87) $\dfrac{x+1}{x+i}$

88) $\dfrac{x^2+1}{x+i}$

89) $\dfrac{x^3+1}{x−i}$

$ \bigstar $ Find the remainder.

90) $(x^4−9x^2+14)÷(x−2)$

91) $(3x^3−2x^2+x−4)÷(x+3)$

92) $(x^4+5x^3−4x−17)÷(x+1)$

93) $(−3x^2+6x+24)÷(x−4)$

94) $(5x^5−4x^4+3x^3−2x^2+x−1)÷(x+6)$

95) $(x^4−1)÷(x−4)$

96) $(3x^3+4x^2−8x+2)÷(x−3)$

97) $(4x^3+5x^2−2x+7)÷(x+2)$

Answers to odd exercises:

85. $1+\dfrac{1+i}{x−i}$

87. $1+\dfrac{1−i}{x+i}$

89. $x^2+ix−1+\dfrac{1−i}{x−i}$

91. $−106$, $f(2) = -106$

93. $0$, $f(4) = 0 $

95. $255$, $f(4) = 255 $

97. $−1$, $f(-2) = -1 $

G: Construct a polynomial from a graph and a given Factor

Exercise $\PageIndex{G}$

$ \bigstar $ 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.

98) Factor is $x^2−x+3$

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