Solutions 17: Zeros of Polynomial Functions

Solutions to Try Its

1. 
f(3)=412f\left(-3\right)=-412


2. The zeros are 2, –2, and –4.

3. There are no rational zeros.

4. The zeros are
-4, 12, and 1.\text{-4, }\frac{1}{2},\text{ and 1}\text{.}


5. 
f(x)=12x3+52x22x+10f\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,
12\frac{1}{2}


17. –2

19. –3

21. 
52,6,6-\frac{5}{2}, \sqrt{6}, -\sqrt{6}


23. 
2,4,322, -4, -\frac{3}{2}


25. 4, –4, –5

27. 
5,3,125, -3, -\frac{1}{2}


29. 
12,1+52,152\frac{1}{2}, \frac{1+\sqrt{5}}{2}, \frac{1-\sqrt{5}}{2}


31. 
32\frac{3}{2}


33. 2, 3, –1, –2

35. 
12,12,2,3\frac{1}{2}, -\frac{1}{2}, 2, -3


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


39. 
34,12-\frac{3}{4}, -\frac{1}{2}


41. 
2,3+2i,32i2, 3+2i, 3 - 2i


43. 
23,1+2i,12i-\frac{2}{3}, 1+2i, 1 - 2i


45. 
12,1+4i,14i-\frac{1}{2}, 1+4i, 1 - 4i


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. 
±5,±1,±52\pm 5, \pm 1, \pm \frac{5}{2}


59. 
±1,±12,±13,±16\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}


61. 
1,12,131, \frac{1}{2}, -\frac{1}{3}


63. 
2,14,322, \frac{1}{4}, -\frac{3}{2}


65. 
54\frac{5}{4}


67. 
f(x)=49(x3+x2x1)f\left(x\right)=\frac{4}{9}\left({x}^{3}+{x}^{2}-x - 1\right)


69. 
f(x)=15(4x3x)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

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