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3.4.4E: Factor Theorem and Remainder Theorem (Exercises)

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section 3.4 exercise

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

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. $$4x^4 - 28x^3 + 61x^2 - 42x + 9 = 4(x - \dfrac{1}{2})^2 (x - 3)^2$$