3.6E: Complex Zeros (Exercises)
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. \(f(x)=x^{4} +5x^{2} -24\)
- Answer
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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: \(x = \pm 2i, \pm i \sqrt{5}\)