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Mathematics LibreTexts

3.6.6E: Complex Zeros (Exercises)

  • Page ID
    13895
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    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. \(f(x)=x^{4} +5x^{2} -24\)

    Answer

    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: \(x = \pm 2i, \pm i \sqrt{5}\)