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Precalculus : Powers and Roots of Complex Numbers

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Example Questions

Example Question #11 : Find The Roots Of Complex Numbers

Solve for $z^{3}-z^{2}+z=0$ (there may be more than one solution).

Possible Answers:

$0,\frac{1}{2},\pm \frac{\sqrt{3i}}{2}$

$0,\frac{1}{2},\pm \sqrt{3i}$

$0,1,\frac{1}{2}$

$0,\frac{1}{2},\pm \frac{\sqrt{3}}{2}$

Correct answer:

$0,\frac{1}{2},\pm \frac{\sqrt{3i}}{2}$

Explanation:

To solve for the roots, just set equal to zero and solve for z using the quadratic formula, which is $\frac{-b\pm \sqrt{b^{2}4ac}}{2a}$

$z^{3}-z^{2}+z\rightarrow z\left ( z^{2}-z+1 \right )$ and now setting both and $z^{2}-z+1$ equal to zero we end up with the answers z=0 and $z=\frac{1\pm 3i}{2}$ and so the correct answer is $0,\frac{1}{2},\pm \frac{2\sqrt{3i}}{2}$ .

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Example Question #12 : Find The Roots Of Complex Numbers

Solve for all possible solutions to the quadratic expression: $m^{2}-6m+12=0$

Possible Answers:

$3\pm \sqrt{3}$

$3\pm 3$

$3\pm 3i$

$3\pm \sqrt{3i}$

Correct answer:

$3\pm \sqrt{3i}$

Explanation:

Solve for complex values of using the quadratic formula: $6\pm\sqrt{36-(4)(1)(12)} = \frac{6\pm\sqrt{-12}}{2}=\frac{6\pm2\sqrt{3}i}{2}=3\pm\sqrt{3}i$ .

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Example Question #43 : Polar Coordinates And Complex Numbers

Solve for $z^{3}-z^{2}+z=0$ (there may be more than one solution).

Possible Answers:

$0,\frac{1}{2},\pm \frac{\sqrt{3i}}{2}$

$0,\frac{1}{2},\pm \sqrt{3i}$

$0,1,\frac{1}{2}$

$0,\frac{1}{2},\pm \frac{\sqrt{3}}{2}$

Correct answer:

$0,\frac{1}{2},\pm \frac{\sqrt{3i}}{2}$

Explanation:

To solve for the roots, just set equal to zero and solve for using the quadratic formula ( $\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$ ): $z^{3}-z^{2}+z\rightarrow z(z^{2}-z+1)$ and now setting both and $z^{2}-z+1$ equal to zero we end up with the answers z=0 and $z=\frac{1\pm 3i}{2}$ .

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Precalculus : Powers and Roots of Complex Numbers

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #11 : Find The Roots Of Complex Numbers

Example Question #12 : Find The Roots Of Complex Numbers

Example Question #43 : Polar Coordinates And Complex Numbers

All Precalculus Resources

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