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
Famous quotes containing the words simple and/or proof:
“No author, without a trial, can conceive of the difficulty of writing a romance about a country where there is no shadow, no antiquity, no mystery, no picturesque and gloomy wrong, nor anything but a commonplace prosperity, in broad and simple daylight, as is happily the case with my dear native land.”
—Nathaniel Hawthorne (1804–1864)
“a meek humble Man of modest sense,
Who preaching peace does practice continence;
Whose pious life’s a proof he does believe,
Mysterious truths, which no Man can conceive.”
—John Wilmot, 2d Earl Of Rochester (1647–1680)