# 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?

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. $$p(x) = x^4 - 5x^3 + x^2 + 5$$, $$c =2$$
 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: $$(x−1)$$
 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. $$p(x)=3x^5 +2x^4 - 15x^3 -10x^2 +12x +8,$$  $$\; c = -\frac{2}{3}$$
 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. $$f(x) = 2x^4+x^3-7x^2-3x+3$$
 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) $$-5x^4+4x^3−19x^2+16x+4=0$$
 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. $$f(x) = -2x^4 - 3x^3 +10x^2 + 12x - 8$$
 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) $$2x^3−3x^2+32x+17=0$$
 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. $$f(x) = x^{4} + 2x^{3} + 6x - 9$$
 79. zeros (odd multiplicity): $$\pm \sqrt{ \frac{1+\sqrt{5} }{2} }$$, 2 imaginary zeros, y-intercept $$(0, 1)$$ 81. zeros (odd multiplicity): $$\{-10, -6, \frac{-5}{2} \}$$; y-intercept: $$(0, 300)$$ 83. zeros (odd multiplicity); $$\{ -1, 1, 3, \frac{-1}{2} \}$$, y-intercept $$(0,3)$$. 85. zeros; $$-4$$ (multiplicity $$2$$), $$1$$ (multiplicity $$1$$), y-intercept $$(0,16)$$. 87.  odd multiplicity zeros: $$\{1, -1\}$$; even multiplicity zero: $$\{ 3 \}$$; y-intercept $$(0, -9)$$. 89. odd multiplicity zero: $$\{ -1 \}$$, even multiplicity zero $$\{ 2 \}$$. y-intercept $$(0, 4)$$. ### 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.

 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 $$x=0.1$$
 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.