In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P.
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It follows from this (and the fundamental theorem of algebra), that if the degree of a real polynomial is odd, it must have at least one real root. That fact can also be proven by using the intermediate value theorem.
Examples and consequences
Corollary on odd-degree polynomials
It follows from the present theorem and the fundamental theorem of algebra that if the degree of a real polynomial is odd, it must have at least one real root.
This can be proved as follows.
This requires some care in the presence of multiple roots; but a complex root and its conjugate do have the same multiplicity (and this lemma is not hard to prove). It can also be worked around by considering only irreducible polynomials; any real polynomial of odd degree must have an irreducible factor of odd degree, which (having no multiple roots) must have a real root by the reasoning above.
This corollary can also be proved directly by using the intermediate value theorem.
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,