## Proof of the Week for 20120723

Theorem: The roots of a polynomial of degree two or higher, with real coefficients occur in complex conjugate pairs.

Proof: Let $z_s=p+iq$ be a solution of $\sum_{k=0}^n a_k z^k = 0$, where $z_s\in\mathbb{C}$, $z^k\in\mathbb{C}$, $a_k$s are real, and $n\ge2$. Let $z_s=p+iq = re^{i\theta}$ so that $z^k = r^ke^{ik\theta}$. Now we have $\sum_{k=0}^n a_k r^ke^{ik\theta} = 0$. Taking the complex conjugate of each side yields, $\sum_{k=0}^n a_k r^ke^{-ik\theta} = 0$, thus $r^ke^{-ik\theta} = \left(re^{-i\theta}\right)^k$ is also a root of the polynomial equation. Noting that $re^{-i\theta} = \bar{z_s}$ completes the proof.