Complex Conjugate Root Theorem - Simple Proof

Simple Proof

One proof of the theorem is as follows:

Consider the polynomial

where all ar are real. Suppose some complex number ζ is a root of P, that is P(ζ) = 0. It needs to be shown that

as well.

If P(ζ) = 0, then

which can be put as

Now

and given the properties of complex conjugation,

Since,

it follows that

That is,

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