
# 3.6.6E: Complex Zeros (Exercises)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

section 3.6 exercise

Simplify each expression to a single complex number.

1. $$\sqrt{-9}$$

2. $$\sqrt{-16}$$

3. $$\sqrt{-6} \sqrt{-24}$$

4. $$\sqrt{-3} \sqrt{-75}$$

5. $$\dfrac{2+\sqrt{-12} }{2}$$

6. $$\dfrac{4+\sqrt{-20} }{2}$$

Simplify each expression to a single complex number.

7. $$\left(3+2i\right)+(5-3i)$$

8. $$\left(-2-4i\right)+\left(1+6i\right)$$

9. $$\left(-5+3i\right)-(6-i)$$

10. $$\left(2-3i\right)-(3+2i)$$

11. $$\left(2+3i\right)(4i)$$

12. $$\left(5-2i\right)(3i)$$

13. $$\left(6-2i\right)(5)$$

14. $$\left(-2+4i\right)\left(8\right)$$

15. $$\left(2+3i\right)(4-i)$$

16. $$\left(-1+2i\right)(-2+3i)$$

17. $$\left(4-2i\right)(4+2i)$$

18. $$\left(3+4i\right)\left(3-4i\right)$$

19. $$\dfrac{3+4i}{2}$$

20. $$\dfrac{6-2i}{3}$$

21. $$\dfrac{-5+3i}{2i}$$

22. $$\dfrac{6+4i}{i}$$

23. $$\dfrac{2-3i}{4+3i}$$

24. $$\dfrac{3+4i}{2-i}$$

Find all of the zeros of the polynomial then completely factor it over the real numbers and completely factor it over the complex numbers.

25. $$f(x)=x^{2} -4x+13$$

26. $$f(x)=x^{2} -2x+5$$

27. $$f(x)=3x^{2} +2x+10$$

28. $$f(x)=x^{3} -2x^{2} +9x-18$$

29. $$f(x)=x^{3} +6x^{2} +6x+5$$

30. $$f(x)=3x^{3} -13x^{2} +43x-13$$

31. $$f(x)=x^{3} +3x^{2} +4x+12$$

32. $$f(x)=4x^{3} -6x^{2} -8x+15$$

33. $$f(x)=x^{3} +7x^{2} +9x-2$$

34. $$f(x)=9x^{3} +2x+1$$

35. $$f(x)=4x^{4} -4x^{3} +13x^{2} -12x+3$$

36. $$f(x)=2x^{4} -7x^{3} +14x^{2} -15x+6$$

37. $$f(x)=x^{4} +x^{3} +7x^{2} +9x-18$$

38. $$f(x)=6x^{4} +17x^{3} -55x^{2} +16x+12$$

39. $$f(x)=-3x^{4} -8x^{3} -12x^{2} -12x-5$$

40. $$f(x)=8x^{4} +50x^{3} +43x^{2} +2x-4$$

41. $$f(x)=x^{4} +9x^{2} +20$$

42. $$f(x)=x^{4} +5x^{2} -24$$

1. 3$$i$$

3. -12

5. $$1 + \sqrt{3} i$$

7. $$8 - i$$

9. $$-11 + 4i$$

11. $$-12 + 8i$$

13. $$30 - 10i$$

15. $$11 + 10i$$

17. 20

19. $$\dfrac{3}{2} + 2i$$

21. $$\dfrac{3}{2} + \dfrac{5}{2} i$$

23. $$-\dfrac{1}{25} - \dfrac{18}{25} i$$

25. $$f(x) = x^2 - 4x + 13 = (x - (2 + 3i))(x - (2 - 3i))$$. Zeros: $$x = 2 \pm 3i$$

27. $$f(x) = 3x^2 + 2x + 10 = 3(x - (-\dfrac{1}{3} + \dfrac{\sqrt{29}}{3}i))(x - (-\dfrac{1}{3} - \dfrac{\sqrt{29}}{3}i))$$. Zeros: $$x = -\dfrac{1}{3} \pm \dfrac{\sqrt{29}}{3}i$$

29. $$f(x) = x^3 + 6x^2 + 6x + 5 = (x + 5) (x^2 + x + 1) = (x + 5)(x - (-\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i))(x - (-\dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i))$$ Zeros: $$x = -5$$, $$x = -\dfrac{1}{2} \pm \dfrac{\sqrt{3}}{2} i$$

31. $$f(x) = x^3 + 3x^2 + 4x + 12 = (x + 3) (x^2 + 4) = (x + 3)(x + 2i)(x - 2i)$$. Zeros: $$x = -3, \pm 2i$$

33. $$f(x) = x^3 + 7x^2 + 9x - 2 = (x + 2)(x - (-\dfrac{5}{2} + \dfrac{\sqrt{29}}{2}))(x - (-\dfrac{5}{2} - \dfrac{\sqrt{29}}{2}))$$ Zeros: $$x = -2$$, $$x = -\dfrac{5}{2} \pm \dfrac{\sqrt{29}}{2}$$

35. $$f(x) = 4x^4 - 4x^3 + 13x^2 - 12x + 3 = (x - \dfrac{1}{2})^2 (4x^2 + 12) = 4(x - \dfrac{1}{2})^2 (x + i \sqrt{3})(x - i \sqrt{3})$$ Zeros: $$x = \dfrac{1}{2}, x = \pm \sqrt{3} i$$

37. $$f(x) = x^4 + x^3 + 7x^2 + 9x - 18 = (x + 2) (x - 1)(x^2 + 9) = (x + 2)(x - 1)(x + 3i)(x - 3i)$$ Zeros: $$x = -2, 1, \pm 3i$$

39. $$f(x) = -3x^4 - 8x^3 - 12x^2 - 12x - 5 = (x + 1)^2 (-3x^2 - 2x - 5) = -3(x + 1)^2 (x - (-\dfrac{1}{3} + \dfrac{\sqrt{14}}{3} i))(x - (-\dfrac{1}{3} - \dfrac{\sqrt{14}}{3} i))$$ Zeors: $$x = -1$$, $$x = -\dfrac{1}{3} \pm \dfrac{\sqrt{14}}{3} i$$

41. $$f(x) = x^4 + 9x^2 + 20 = (x^2 + 4)(x^2 + 5) = (x - 2i)(x + 2i)(x - i\sqrt{5}) (x + i\sqrt{5})$$ Zeros: $$x = \pm 2i, \pm i \sqrt{5}$$