3.5E: Real Zeros of Polynomials (Exercises)

section 3.5 exercise

For each of the following polynomials, use Cauchy’s Bound to find an interval containing all the real zeros, then use Rational Roots Theorem to make a list of possible rational zeros.

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

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

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

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

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

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

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

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

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

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

Find the real zeros of each polynomial.

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

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

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

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

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

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

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

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

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

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

21. $f(x)=9x^{3} -5x^{2} -x$

22. $f(x)=6x^{4} -5x^{3} -9x^{2}$

23. $f(x)=x^{4} +2x^{2} -15$

24. $f(x)=x^{4} -9x^{2} +14$

25. $f(x)=3x^{4} -14x^{2} -5$

26. $f(x)=2x^{4} -7x^{2} +6$

27. $f(x)=x^{6} -3x^{3} -10$

28. $f(x)=2x^{6} -9x^{3} +10$

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

30. $f(x)=2x^{5} +3x^{4} -18x-27$

31. $f(x)=x^{5} -60x^{3} -80x^{2} +960x+2304$

32. $f(x)=25x^{5} -105x^{4} +174x^{3} -142x^{2} +57x-9$

Answer

1. All the real zeros lie in the interval [-7, 7]

-Possible rational zeros are $\pm 1, \pm 2, \pm 3$

3. All of the real zeros lie in the interval [-13, 13]

-Possible rational zeros are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$

5. All of the real zeors lie in the interval [-8, 8]

-Possible rational zeors are $\pm 1, \pm 7$

7. All of the real zeros lie in the interval [-3, 3]

-Possible rational zeros are $\pm \dfrac{1}{17}, \pm \dfrac{2}{17}, \pm \dfrac{5}{17}, \pm \dfrac{10}{17}, \pm 1, \pm 2, \pm 5, \pm 10$

9. All of the real zeros lie in the interval $[-\dfrac{14}{3}, \dfrac{14}{3}]$

-Possible rational zeros are $\pm \dfrac{1}{3}, \pm \dfrac{2}{3}, \pm \dfrac{5}{3}, \pm \dfrac{10}{3}, \pm 1, \pm 2, \pm 5, \pm 10$

11. $x = -2, x = 1, x = 3$ (each has mult. 1)

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

15. $x = 7$ (mult. 1)

17. $x = \dfrac{5}{17}, x = \pm \sqrt{2}$ (each has mult. 1)

19. $x = -2, x = \dfrac{3 \pm \sqrt{69}} {6}$ (each has mult. 1)

21. $x = 0, x = \dfrac{5 \pm \sqrt{61}}{18}$ (each has mult. 1)

23. $x = \pm \sqrt{3}$ (each has mult. 1)

25. $x = \pm \sqrt{5}$ (each has mult. 1)

27. $x = \sqrt[3]{-2} = -\sqrt[3]{2}, x = \sqrt[3]{5}$ (each has mult. 1)

29. $x = 2, x = \pm \sqrt{2}$ (each has mult. 1)

31. $x = -4$ (mult. 3), $x = 6$ (mult. 2)