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_ks 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.

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