If $3 – 5i$ is a square root of $z$, find the other root.
Well, I was under the impression that only the sign in front of the imaginary part would change so the other root would be $3 + 5i$.
However, when I solve for the complex root using de Moivre's theorem, then I get $-3 + 5i$.
Both the real and imaginary parts have changed signs – this seems to go against what I thought was the complex conjugate root theorem?
Best Answer
If we apply DeMoivre's Theorem, the second root will be a $180$ degree rotation around the pole. So the other root is in fact, $-3 + 5i$ (you're right!). You are confusing the usage of the complex conjugate root theorem, which only applies to polynomials with real coefficients, not square roots of complex numbers in the complex plane.