3.5E: Real Zeros of Polynomials (Exercises)
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)