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Mathematics LibreTexts

3.6E: Complex Zeros (Exercises)

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  • Page ID
    13895
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    Simplify each expression to a single complex number. \[1. \sqrt{-9} 2. \sqrt{-16} 3. \sqrt{-6} \sqrt{-24}\] \[4. \sqrt{-3} \sqrt{-75} 5. \frac{2+\sqrt{-12} }{2} 6. \frac{4+\sqrt{-20} }{2}\]

    Simplify each expression to a single complex number. \[7. \left(3+2i\right)+(5-3i) 8. \left(-2-4i\right)+\left(1+6i\right)\] \[9. \left(-5+3i\right)-(6-i) 10. \left(2-3i\right)-(3+2i)\] \[11. \left(2+3i\right)(4i) 12. \left(5-2i\right)(3i)\] \[13. \left(6-2i\right)(5) 14. \left(-2+4i\right)\left(8\right)\] \[15. \left(2+3i\right)(4-i) 16. \left(-1+2i\right)(-2+3i)\] \[17. \left(4-2i\right)(4+2i) 18. \left(3+4i\right)\left(3-4i\right)\] \[19. \frac{3+4i}{2} 20. \frac{6-2i}{3}\] \[21. \frac{-5+3i}{2i} 22. \frac{6+4i}{i}\] \[23. \frac{2-3i}{4+3i} 24. \frac{3+4i}{2-i}\]

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

    \[25. f(x)=x^{2} -4x+13 26. f(x)=x^{2} -2x+5\] \[27. f(x)=3x^{2} +2x+10 28. f(x)=x^{3} -2x^{2} +9x-18\] \[29. f(x)=x^{3} +6x^{2} +6x+5 30. f(x)=3x^{3} -13x^{2} +43x-13\] \[31. f(x)=x^{3} +3x^{2} +4x+12 32. f(x)=4x^{3} -6x^{2} -8x+15\] \[33. f(x)=x^{3} +7x^{2} +9x-2 34. f(x)=9x^{3} +2x+1\] \[35. f(x)=4x^{4} -4x^{3} +13x^{2} -12x+3 36. f(x)=2x^{4} -7x^{3} +14x^{2} -15x+6\] \[37. f(x)=x^{4} +x^{3} +7x^{2} +9x-18 38. f(x)=6x^{4} +17x^{3} -55x^{2} +16x+12\] \[39. f(x)=-3x^{4} -8x^{3} -12x^{2} -12x-5 40. f(x)=8x^{4} +50x^{3} +43x^{2} +2x-4\] \[41. f(x)=x^{4} +9x^{2} +20 42. f(x)=x^{4} +5x^{2} -24\]

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    3.7 Rational Functions