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# 3.4E: Factor Theorem and Remainder Theorem (Exercises)

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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

$209$

3.5 Real Zeros of Polynomials