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Mathematics LibreTexts

3.4E: Factor Theorem and Remainder Theorem (Exercises)

  • Page ID
    13892
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    Use polynomial long division to perform the indicated division.

    1. \(\left(4x^{2} +3x-1\right)\div (x-3)\)

    2. \(\left(2x^{3} -x+1\right)\div \left(x^{2} +x+1\right)\) 

    3. \(\left(5x^{4} -3x^{3} +2x^{2} -1\right)\div \left(x^{2} +4\right)\)

    4. \(\left(-x^{5} +7x^{3} -x\right)\div \left(x^{3} -x^{2} +1\right)\) 

    5. \(\left(9x^{3} +5\right)\div \left(2x-3\right)\)

    6. \(\left(4x^{2} -x-23\right)\div \left(x^{2} -1\right)\)

    Use synthetic division to perform the indicated division.

    7. \(\left(3x^{2} -2x+1\right)\div \left(x-1\right)\)

    8. \(\left(x^{2} -5\right)\div \left(x-5\right)\) 

    9. \(\left(3-4x-2x^{2} \right)\div \left(x+1\right)\)

    10. \(\left(4x^{2} -5x+3\right)\div \left(x+3\right)\) 

    11. \(\left(x^{3} +8\right)\div \left(x+2\right)\)

    12. \(\left(4x^{3} +2x-3\right)\div \left(x-3\right)\) 

    13. \(\left(18x^{2} -15x-25\right)\div \left(x-\frac{5}{3} \right)\) 

    14. \(\left(4x^{2} -1\right)\div \left(x-\frac{1}{2} \right)\) 

    15. \(\left(2x^{3} +x^{2} +2x+1\right)\div \left(x+\frac{1}{2} \right)\)

    16. \(\left(3x^{3} -x+4\right)\div \left(x-\frac{2}{3} \right)\) 

    17. \(\left(2x^{3} -3x+1\right)\div \left(x-\frac{1}{2} \right)\)

    18. \(\left(4x^{4} -12x^{3} +13x^{2} -12x+9\right)\div \left(x-\frac{3}{2} \right)\) 

    19. \(\left(x^{4} -6x^{2} +9\right)\div \left(x-\sqrt{3} \right)\) 

    20. \(\left(x^{6} -6x^{4} +12x^{2} -8\right)\div \left(x+\sqrt{2} \right)\)

    Below you are given a polynomial and one of its zeros. Use the techniques in this section to find the rest of the real zeros and factor the polynomial.

    21. \(x^{3} -6x^{2} +11x-6,\; \; c=1\)

    22. \(x^{3} -24x^{2} +192x-512,\; \; c=8\) 

    23. \(3x^{3} +4x^{2} -x-2,\; \; c=\frac{2}{3}\)

    24. \(2x^{3} -3x^{2} -11x+6,\; \; c=\frac{1}{2}\) 

    25. \(x^{3} +2x^{2} -3x-6,\; \; c=-2\)

    26. \(2x^{3} -x^{2} -10x+5,\; \; c=\frac{1}{2}\) 

    27. \(4x^{4} -28x^{3} +61x^{2} -42x+9\), \(c=\frac{1}{2}\) is a zero of multiplicity 2

    28. \(x^{5} +2x^{4} -12x^{3} -38x^{2} -37x-12\), \(c=-1\) is a zero of multiplicity 3