Solutions 17: Zeros of Polynomial Functions

Solutions to Try Its

1. [latex]f\left(-3\right)=-412[/latex]

2. The zeros are 2, –2, and –4.

3. There are no rational zeros.

4. The zeros are [latex]\text{-4, }\frac{1}{2},\text{ and 1}\text{.}[/latex]

5. [latex]f\left(x\right)=-\frac{1}{2}{x}^{3}+\frac{5}{2}{x}^{2}-2x+10[/latex]

6. There must be 4, 2, or 0 positive real roots and 0 negative real roots. The graph shows that there are 2 positive real zeros and 0 negative real zeros.

7. 3 meters by 4 meters by 7 meters

Solutions to Odd-Numbered 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. Polynomial functions can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.

7. –106

9. 0

11. 255

13. –1

15. –2, 1, [latex]\frac{1}{2}[/latex]

17. –2

19. –3

21. [latex]-\frac{5}{2}, \sqrt{6}, -\sqrt{6}[/latex]

23. [latex]2, -4, -\frac{3}{2}[/latex]

25. 4, –4, –5

27. [latex]5, -3, -\frac{1}{2}[/latex]

29. [latex]\frac{1}{2}, \frac{1+\sqrt{5}}{2}, \frac{1-\sqrt{5}}{2}[/latex]

31. [latex]\frac{3}{2}[/latex]

33. 2, 3, –1, –2

35. [latex]\frac{1}{2}, -\frac{1}{2}, 2, -3[/latex]

37. [latex]-1, -1, \sqrt{5}, -\sqrt{5}[/latex]

39. [latex]-\frac{3}{4}, -\frac{1}{2}[/latex]

41. [latex]2, 3+2i, 3 - 2i[/latex]

43. [latex]-\frac{2}{3}, 1+2i, 1 - 2i[/latex]

45. [latex]-\frac{1}{2}, 1+4i, 1 - 4i[/latex]

47. 1 positive, 1 negative
Graph of f(x)=x^4-x^2-1.

49. 3 or 1 positive, 0 negative
Graph of f(x)=x^3-2x^2+x-1.

51. 0 positive, 3 or 1 negative
Graph of f(x)=2x^3+37x^2+200x+300.

53. 2 or 0 positive, 2 or 0 negative
Graph of f(x)=2x^4-5x^3-5x^2+5x+3.

55. 2 or 0 positive, 2 or 0 negative
Graph of f(x)=10x^4-21x^2+11.

57. [latex]\pm 5, \pm 1, \pm \frac{5}{2}[/latex]

59. [latex]\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}[/latex]

61. [latex]1, \frac{1}{2}, -\frac{1}{3}[/latex]

63. [latex]2, \frac{1}{4}, -\frac{3}{2}[/latex]

65. [latex]\frac{5}{4}[/latex]

67. [latex]f\left(x\right)=\frac{4}{9}\left({x}^{3}+{x}^{2}-x - 1\right)[/latex]

69. [latex]f\left(x\right)=-\frac{1}{5}\left(4{x}^{3}-x\right)[/latex]

71. 8 by 4 by 6 inches

73. 5.5 by 4.5 by 3.5 inches

75. 8 by 5 by 3 inches

77. Radius = 6 meters, Height = 2 meters

79. Radius = 2.5 meters, Height = 4.5 meters