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,
Read more about this topic: Complex Conjugate Root Theorem
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