Solutions 17: Zeros of Polynomial Functions

Solutions to Try Its

1.
$f\left(-3\right)=-412$

2. The zeros are 2, –2, and –4.

3. There are no rational zeros.

4. The zeros are
$\text{-4, }\frac{1}{2},\text{ and 1}\text{.}$

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

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,
$\frac{1}{2}$

17. –2

19. –3

21.
$-\frac{5}{2}, \sqrt{6}, -\sqrt{6}$

23.
$2, -4, -\frac{3}{2}$

25. 4, –4, –5

27.
$5, -3, -\frac{1}{2}$

29.
$\frac{1}{2}, \frac{1+\sqrt{5}}{2}, \frac{1-\sqrt{5}}{2}$

31.
$\frac{3}{2}$

33. 2, 3, –1, –2

35.
$\frac{1}{2}, -\frac{1}{2}, 2, -3$

37.
$-1, -1, \sqrt{5}, -\sqrt{5}$

39.
$-\frac{3}{4}, -\frac{1}{2}$

41.
$2, 3+2i, 3 - 2i$

43.
$-\frac{2}{3}, 1+2i, 1 - 2i$

45.
$-\frac{1}{2}, 1+4i, 1 - 4i$

47. 1 positive, 1 negative

49. 3 or 1 positive, 0 negative

51. 0 positive, 3 or 1 negative

53. 2 or 0 positive, 2 or 0 negative

55. 2 or 0 positive, 2 or 0 negative

57.
$\pm 5, \pm 1, \pm \frac{5}{2}$

59.
$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}$

61.
$1, \frac{1}{2}, -\frac{1}{3}$

63.
$2, \frac{1}{4}, -\frac{3}{2}$

65.
$\frac{5}{4}$

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

69.
$f\left(x\right)=-\frac{1}{5}\left(4{x}^{3}-x\right)$

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