
# 3.5E: Real Zeros of Polynomials (Exercises)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

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