3.4E: Factor Theorem and Remainder Theorem (Exercises)
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. \(\left(18x^{2} -15x-25\right)\div \left(x-\dfrac{5}{3} \right)\)
14. \(\left(4x^{2} -1\right)\div \left(x-\dfrac{1}{2} \right)\)
15. \(\left(2x^{3} +x^{2} +2x+1\right)\div \left(x+\dfrac{1}{2} \right)\)
16. \(\left(3x^{3} -x+4\right)\div \left(x-\dfrac{2}{3} \right)\)
17. \(\left(2x^{3} -3x+1\right)\div \left(x-\dfrac{1}{2} \right)\)
18. \(\left(4x^{4} -12x^{3} +13x^{2} -12x+9\right)\div \left(x-\dfrac{3}{2} \right)\)
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. \(3x^{3} +4x^{2} -x-2,\; \; c=\dfrac{2}{3}\)
24. \(2x^{3} -3x^{2} -11x+6,\; \; c=\dfrac{1}{2}\)
25. \(x^{3} +2x^{2} -3x-6,\; \; c=-2\)
26. \(2x^{3} -x^{2} -10x+5,\; \; c=\dfrac{1}{2}\)
27. \(4x^{4} -28x^{3} +61x^{2} -42x+9\), \(c=\dfrac{1}{2}\) 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
- Answer
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1. \(4x^2 + 3x - 1 = (x - 3) (4x + 15) + 44\)
3. \(5x^4 - 3x^3 + 2x^2 - 1 = (x^2 + 4) (5x^2 - 3x - 18) + (12x + 71)\)
5. \(9x^3 + 5 = (2x - 3) (\dfrac{9}{2}x^2 + \dfrac{27}{4} x + \dfrac{81}{8}) + \dfrac{283}{8}\)
7. \((3x^2 - 2x + 1) = (x - 1)(3x + 1) +2\)
9. \((3 - 4x - 2x^2) = (x + 1) (-2x - 2) + 5\)
11. \((x^3 + 8) = (x + 2)(x^2 - 2x + 4) + 0\)
13. \((18x^2 - 15x - 25) = (x - \dfrac{5}{3})(18x + 15) + 0\)
15. \((2x^3 +x^2 + 2x + 1) = (x + \dfrac{1}{2})(2x^2 + 2) + 0\)
17. \((2x^3 - 3x + 1) = (x - \dfrac{1}{2})(2x^2 + x - \dfrac{5}{2}) - \dfrac{1}{4}\)
19. \((x^4 - 6x^2 + 9) = (x - \sqrt{3}) (x^3 + \sqrt{3}x^2 - 3x - 3\sqrt{3}) + 0\)
21. \(x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2) (x - 3)\)
23. \(3x^3 + 4x^2 - x - 2 = 3(x - \dfrac{2}{3})(x + 1)^2\)
25. \(x^3 + 2x^2 - 3x - 6 = (x + 2) (x + \sqrt{3}) (x - \sqrt{3})\)
27. \(4x^4 - 28x^3 + 61x^2 - 42x + 9 = 4(x - \dfrac{1}{2})^2 (x - 3)^2\)