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Mathematics LibreTexts

3.5E: Real Zeros of Polynomials (Exercises)

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    13893
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    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. \[11. f(x)=x^{3} -2x^{2} -5x+6 12. f(x)=x^{4} +2x^{3} -12x^{2} -40x-32\] \[13. f(x)=x^{4} -9x^{2} -4x+12 14. f(x)=x^{3} +4x^{2} -11x+6\] \[15. f(x)=x^{3} -7x^{2} +x-7 16. f(x)=-2x^{3} +19x^{2} -49x+20\] \[17. f(x)=-17x^{3} +5x^{2} +34x-10 18. f(x)=36x^{4} -12x^{3} -11x^{2} +2x+1\] \[19. f(x)=3x^{3} +3x^{2} -11x-10 20. f(x)=2x^{4} +x^{3} -7x^{2} -3x+3\] \[21. f(x)=9x^{3} -5x^{2} -x 22. f(x)=6x^{4} -5x^{3} -9x^{2}\] \[23. f(x)=x^{4} +2x^{2} -15 24. f(x)=x^{4} -9x^{2} +14\] \[25. f(x)=3x^{4} -14x^{2} -5 26. f(x)=2x^{4} -7x^{2} +6\] \[27. f(x)=x^{6} -3x^{3} -10 28. f(x)=2x^{6} -9x^{3} +10\] \[29. f(x)=x^{5} -2x^{4} -4x+8 30. f(x)=2x^{5} +3x^{4} -18x-27\] \[31. f(x)=x^{5} -60x^{3} -80x^{2} +960x+2304\] \[32. f(x)=25x^{5} -105x^{4} +174x^{3} -142x^{2} +57x-9\]

    \[219\]