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Mathematics LibreTexts

3.5E: Real Zeros of Polynomials (Exercises)

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