3.6e: Exercises - Zeroes of Polynomial Functions

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A: Concepts

Exercise $\PageIndex{A}$: Concepts

1) Describe a use for the Remainder Theorem.

2) Explain why the Rational Zero Theorem does not guarantee finding zeros of a polynomial function.

3) What is the difference between rational and real zeros?

4) If Descartes’ Rule of Signs reveals a $0$ or $1$ change of signs, what specific conclusion can be drawn?

5) If synthetic division reveals a zero, why should we try that value again as a possible solution?

Answers to odd exercises:

1. The theorem can be used to evaluate a polynomial.

3. Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.

5. Polynomials can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.

B: Use the Remainder Theorem to Evaluate a Polynomial

Exercise $\PageIndex{B}$: Use the Remainder Theorem

$ \bigstar $ Use synthetic division to evaluate $p(c)$ and write $p(x)$ in the form $p(x) = (x-c) q(x) +r$.

6. $p(x) = 2x^2 - x + 1$, $c = 4$

7. $p(x) = 4x^2-33x-180$, $c = 12$

8. $p(x) = 2x^3 - x + 6$, $c=-3$

9. $p(x) = x^3+2x^2+3x+4$, $c =-1$

10. $p(x) =3x^3-6x^2+4x-8$, $c=2$

11. $p(x) = 8x^3+12x^2+6x+1$, $c =-\frac{1}{2}$

12. $p(x) = 2x^4 +x^3- 4x^2+10x-7$, $c=\frac{3}{2}$

13. $p(x) = x^4 - 3x^3 - 20x^2 - 24x - 8$, $c =7$

14. $p(x) = x^4 - 5x^2 - 8x -12$, $c=3$

15. $p(x) = x^4 - 5x^3 + x^2 + 5$, $c =2$

Answers to odd exercises:

7. $p(12) =0$, $p(x) = (x-12)(4x+15) $

9. $p(-1)=2$, $p(x) = (x+1)(x^2 + x+2) + 2 $

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

13. $p(7)=216$, $p(x) = (x-7)(x^3+4x^2 +8 x+32) + 216 $

15. $p(2)=-15$, $p(x) = (x-2)(x^3-3x^2 -5x -10) -15 $

C: Given one zero or factor, find all Real Zeros, and factor a polynomial

Exercise $\PageIndex{C}$: Use the Factor Theorem given one zero or factor

$ \bigstar $ Given a polynomial and one of its factors, find the rest of the real zeros and write the polynomial as a product of linear and irreducible quadratic factors.

17) $f(x)=2x^3+x^2−5x+2;$ Factor: $ ( x+2) $

18) $f(x)=3x^3+x^2−20x+12;$ Factor: $ ( x+3)$

19) $f(x)=2x^3+3x^2+x+6;$ Factor: $ (x+2)$

20) $f(x)=−5x^3+16x^2−9;$ Factor: $ (x−3)$

21) $f(x)=x^3+3x^2+4x+12;$ Factor: $ (x+3)$

22) $f(x)=4x^3−7x+3;$ Factor: $ (x−1)$

23) $f(x)=2x^3+5x^2−12x−30;$ Factor: $ (2x+5)$

24) $f(x)=2x^3−9x^2+13x−6;$ Factor: $ (x−1) $

Answers to odd exercises:

17. $−2, 1, \frac{1}{2}$; $ f(x)=(x+2)(x-1)(2x-1) $

19. $−2$; $ f(x)=(x+2)(2x^2-x+3) $

21. $−3$; $ f(x)=(x+3)(x^2+4) $

23. $−\frac{5}{2},\; \sqrt{6},\; −\sqrt{6}; $ $f(x)=(2x+5)(x-\sqrt{6})(x+\sqrt{6})$

$ \bigstar $ Given a polynomial and $c$, one of its zeros, find the rest of the real zeros and write the polynomial as a product of linear and irreducible quadratic factors. It is possible some factors are repeated.

25. $p(x)=x^{3} - 24x^{2} + 192x - 512, \;\; c = 8$

26. $p(x)=3x^{3} + 4x^{2} - x - 2, \;\; c = \frac{2}{3}$

27. $p(x)=2x^3-3x^2-11x+6, \;\; c=\frac{1}{2}$

28. $p(x)=x^3+2x^2-3x-6, \;\; c = -2$

29. $p(x)=2x^3-x^2-10x+5, \;\; c=\frac{1}{2}$

30. $p(x)=4x^{4} - 28x^{3} + 61x^{2} - 42x + 9,\; c = \frac{1}{2}$

31. $p(x)=x^5+2x^4-12x^3-38x^2-37x-12,$ $\; c=-1$

32. $p(x)=2x^5 +7x^4 - 18x^2 - 8x +8,$ $\; c = \frac{1}{2}$

33. $p(x)=3x^5 +2x^4 - 15x^3 -10x^2 +12x +8,$ $\; c = -\frac{2}{3}$

Answers to odd exercises:

25. zeros: $8$; $p(x)= (x - 8)^{3}$

27. zeros: $ \frac{1}{2}, -2, 3 $; $p(x)= (2x-1)(x+2)(x-3)$

29. zeros: $ \frac{1}{2}, \pm \sqrt{5} $; $p(x)= (2x-1)(x+\sqrt{5})(x-\sqrt{5})$

31. zeros: $ -1,$ $-3,$ $4$; $p(x)= (x+1)^3(x+3)(x-4)$

33. zeros: $ -2,\; -1,\; -\frac{2}{3},\; 1,\; 2 \\ $;
$ \quad$ $p(x)= (x+2)(x+1)(x-1)(x-2)(3x+2)$

D: List all Possible Rational Zeros

Exercise $\PageIndex{D}$: Use the Rational Zero Theorem

$ \bigstar $ Use the Rational Zeros Theorem to list all possible rational zeros for each given function.

35) $f(x)=2x^3+3x^2−8x+5$

36) $f(x)=3x^3+5x^2−5x+4$

37) $f(x)=6x^4−10x^2+13x+1$

38) $f(x)=4x^5−10x^4+8x^3+x^2−8$

39. $f(x) = x^{3} - 2x^{2} - 5x + 6$

40. $f(x) = x^{4} + 2x^{3} - 12x^{2} - 40x - 32$

41. $f(x) = x^{3} - 7x^{2} + x - 7$

42. $f(x) = x^{3} + 4x^{2} - 11x + 6$

43. $f(x) = x^{4} - 9x^{2} - 4x + 12$

44. $f(x) = -2x^{3} + 19x^{2} - 49x + 20$

45. $f(x) = -17x^{3} + 5x^{2} + 34x - 10$

46. $f(x) = 36x^{4} - 12x^{3} - 11x^{2} + 2x + 1$

47. $f(x) = 3x^{3} + 3x^{2} - 11x - 10$

48. $f(x) = 2x^4+x^3-7x^2-3x+3$

Answers to odd exercises:

35. $±5, ±1, ± \frac{1}{2}, ± \frac{5}{2}$

37. $±1, ±\frac{1}{2}, ±\frac{1}{3}, ±\frac{1}{6}$

39. $\pm 1$, $\pm 2$, $\pm 3$, $\pm 6$ $\qquad \qquad$ 41. $\pm 1$, $\pm 7$

43. $\pm 1$, $\pm 2$, $\pm 3$, $\pm 4$, $\pm 6$, $\pm 12$

45. $\pm 1$, $\pm 2$, $\pm 5$, $\pm 10$, $\pm \frac{1}{17}$,$\pm \frac{2}{17}$,$\pm \frac{5}{17}$,$\pm \frac{10}{17}$

47. $\pm 1$, $\pm 2$, $\pm 5$, $\pm 10$, $\pm \frac{1}{3}$,$\pm \frac{2}{3}$,$\pm \frac{5}{3}$,$\pm \frac{10}{3}$

E: Find all Zeros that are Rational

Exercise $\PageIndex{E}$: Find all zeros that are rational

$ \bigstar $ Use the Rational Zero Theorem to find all real number zeros.

49) $x^3−3x^2−10x+24=0$

50) $2x^3+7x^2−10x−24=0$

51) $x^3+2x^2−9x−18=0$

52) $x^3+5x^2−16x−80=0$

53) $x^3−3x^2−25x+75=0$

54) $2x^3−3x^2−32x−15=0$

55) $2x^3+x^2−7x−6=0$

56) $2x^3−3x^2−x+1=0$

57) $3x^3−x^2−11x−6=0$

58) $x^4−2x^3−7x^2+8x+12=0$

59) $4x^3−3x+1=0$

60) $4x^4+4x^3−25x^2−x+6=0$

61) $x^4+2x^3−9x^2−2x+8=0$

62) $x^4+2x^3−4x^2−10x−5=0$

63) $-5x^4+4x^3−19x^2+16x+4=0$

Answers to odd exercises:

49. $-3,\; 2,\; 4$

51. $-2,\; 3,\; −3$

53. $3, −5, 5 $

55. $2,\; -1,\; -\frac{3}{2}$

57. $ -\frac{2}{3} ,\; \frac{1 \pm \sqrt{13}}{2} $

59. $ -1,\; \frac{1}{2} $

61. $1,\; 2,\; −1,\; −4$

63. $−\frac{1}{5},\; 1 $

$ \bigstar $ Find the real zeros of the polynomial. State the multiplicity of each real zero.

65. $f(x) = x^{3} - 2x^{2} - 5x + 6$

66. $f(x) = x^{3} + 4x^{2} - 11x + 6$

67. $f(x) = x^{4} - 9x^{2} - 4x + 12$

68. $f(x) = -17x^{3} + 5x^{2} + 34x - 10$

69. $f(x) = 36x^{4} - 12x^{3} - 11x^{2} + 2x + 1$

70. $f(x) = 2x^4+x^3-7x^2-3x+3$

71. $f(x) = 2x^{3} + 7x^{2} +4x - 4$

72. $f(x) = -2x^4 - 3x^3 +10x^2 + 12x - 8$

Answers to odd exercises:

65. $x = 1$ (mult. 1), $x = 3$ (mult. 1), $x = -2$ (mult. 1)

67. $x = -2$ (mult. 2), $x = 1$ (mult. 1), $x = 3$ (mult. 1)

69. $x = \frac{1}{2}$ (mult. 2), $ x = -\frac{1}{3}$ (mult. 2)

71. $x = -2$ (mult. 2), $x = \frac{1}{2}$ (mult. 1)

F: Find all zeros (both real and imaginary)

Exercise $\PageIndex{F}$: Find all zeros

$ \bigstar $ Use the Rational Zero Theorem to find all complex solutions (real and non-real).

72) $x^3+x^2+x+1=0$

73) $x^3−8x^2+25x−26=0$

74) $x^3+13x^2+57x+85=0$

75) $3x^3−4x^2+11x+10=0$

76) $x^4+2x^3+22x^2+50x−75=0$

77) $2x^3−3x^2+32x+17=0$

Answers to odd exercises:

73. $2, 3+2i, 3−2i$

75. $−\dfrac{2}{3}, 1+2i, 1−2i$

77. $−\dfrac{1}{2}, 1+4i, 1−4i$

G: Find all zeros and sketch

Exercise $\PageIndex{G}$: Find all zeros and sketch

$ \bigstar $ Determine the end behaviour, all the real zeros, their multiplicity, and y-intercept. Sketch the function. (Use synthetic division to find a rational zero. Use the quotient to find the next zero).

78) $f(x)=x^3−1$

79) $f(x)=x^4−x^2−1$

80) $f(x)=x^3−2x^2−5x+6$

81) $f(x)=2x^3+37x^2+200x+300$

82) $f(x)=x^4+2x^3−12x^2+14x−5$

83) $f(x)=2x^4−5x^3−5x^2+5x+3$

84) $f(x)=x^3−2x^2−16x+32$

85) $f(x)=-x^3−7x^2-8x+16$

86. $f(x) = -x^4 - 4x^3+3x^2 +10x -8$

87. $f(x) = x^{4} - 6x^{3} + 8x^{2} + 6x - 9$

88. $f(x) = x^{4} + 4x^{3} - 5x^{2} - 36x - 36$

89. $f(x) = x^{5} - x^{4} - 5x^{3} + x^{2} + 8x + 4$

90. $f(x) = x^{4} + 2x^{3} + 6x - 9$

Answers to odd exercises:

79. zeros (odd multiplicity): $ \pm \sqrt{ \frac{1+\sqrt{5} }{2} }$, 2 imaginary zeros, y-intercept $ (0, 1) $

$CNX_PreCalc_Figure_03_06_202.jpg$

81. zeros (odd multiplicity): $ \{-10, -6, \frac{-5}{2} \} $; y-intercept: $ (0, 300) $

$CNX_PreCalc_Figure_03_06_206 (1).jpg$

83. zeros (odd multiplicity); $ \{ -1, 1, 3, \frac{-1}{2} \} $, y-intercept $ (0,3) $.

$CNX_PreCalc_Figure_03_06_208.jpg$

85. zeros; $-4$ (multiplicity $2$), $1$ (multiplicity $1$), y-intercept $ (0,16) $.

$3.6E #85.png$

87. odd multiplicity zeros: $ \{1, -1\} $; even multiplicity zero: $ \{ 3 \} $; y-intercept $ (0, -9) $.

$3.6E #87.png$

89. odd multiplicity zero: $ \{ -1 \} $, even multiplicity zero $ \{ 2 \} $. y-intercept $ (0, 4) $.

$3.6E #89.png$

H: Given zeros, construct a polynomial function

Exercise $\PageIndex{H}$: Given zeros, construct a polynomial function

$ \bigstar $ Construct a polynomial function of least degree possible using the given information. You may leave the polynomial in factored form.

91) A lowest degree polynomial with real coefficients and zero $ 3i $

92) A lowest degree polynomial with rational coefficients and zeros: $ 2 $ and $ \sqrt{6} $

93) A lowest degree polynomial with integer coefficients and Real roots: $–1$ (with multiplicity $2$), and $1$.

94) A lowest degree polynomial with integer coefficients and Real roots: $–2$, and $\frac{1}{2}$ (with multiplicity $2$)

95) A lowest degree polynomial with integer coefficients and Real roots:$−\frac{1}{2}, 0,\frac{1}{2}$

96) A lowest degree polynomial with integer coefficients and Real roots: $–4, –1, 1, 4$

97) A lowest degree polynomial with integer coefficients and Real roots: $–1, 1, 3$

98. A lowest degree polynomial with real coefficients and zeros: $-2 $ and $ -5i $

99. A lowest degree polynomial with real coefficients and zeros: $4 $ and $ 2i $.

100. The solutions to $p(x) = 0$ are $x = \pm 3$ and $x=6$. The leading term of $p(x)$ is $7x^4$.
$\qquad$The point $(-3,0)$ is a local minimum on the graph of $y=p(x)$.

101. The solutions to $p(x) =0$ are $x = \pm 3$, $x=-2$, and $x=4$, The leading term of $p(x)$ is $-x^5$.
$\qquad$The point $(-2, 0)$ is a local maximum on the graph of $y=p(x)$.

102. $p$ is degree 4. as $x \rightarrow \infty$, $p(x) \rightarrow -\infty$ $p$ has exactly three $x$-intercepts: $(-6,0)$, $(1,0)$ and $(117,0)$.
$\qquad$The graph of $y=p(x)$ crosses through the $x$-axis at $(1,0)$.

103. Find a quadratic polynomial with integer coefficients which has $x = \dfrac{3}{5} \pm \dfrac{\sqrt{29}}{5}$ as its real zeros.

Answers to odd exercises:

91. $f(x)=(x^2+9)$

93. $f(x)=(x+1)^2(x-1)$

95. $f(x)=x(2x+1)(2x-1)$

97. $ f(x) = (x+1)(x-1)(x-3) $

99. $p(x)= (x-4)(x-2i)(x+2i)=x^3-4x^2+4x-16$

101. $p(x) = -(x + 2)^{2}(x - 3)(x + 3)(x - 4)$

103. $p(x) = 5x^{2} - 6x - 4$

I: Use Intermediate Value Theorem

Exercise $\PageIndex{I}$: Intermediate Value Theorem

$ \bigstar $ Use the Intermediate Value Theorem to confirm the polynomial $f$ has at least one zero within the given interval.

104) $f(x)=x^3−9x$, between $x=−4$ and $x=−2$.

105) $f(x)=x^3−9x$, between $x=2$ and $x=4$.

106) $f(x)=x^5−2x$, between $x=1$ and $x=2$.

107) $f(x)=−x^4+4$, between $x=1$ and $x=3$.

108) $f(x)=−2x^3−x$, between $x=–1$ and $x=1$.

109) $f(x)=x^3−100x+2$, between $x=0.01$ and $x=0.1$

Answers to odd exercises:

105. $f(2)=–10,\; f(4)=28$.
$\qquad$Sign change confirms.

107. $f(1)=3,\; f(3)=–77.$
$\qquad$Sign change confirms.

109. $f(0.01)=1.000001,\; f(0.1)=–7.999$.
$\qquad$Sign change confirms.