Complex conjugate root theorem


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 being real numbers, then its complex conjugate a − bi is also a root of P.
It follows from this that, if the degree of a real polynomial is odd, it must have at least one real root. That fact can also be proved by using the intermediate value theorem.

Examples and consequences

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.
  • Since non-real complex roots come in conjugate pairs, there are an even number of them;
  • But a polynomial of odd degree has an odd number of roots ;
  • Therefore some of them must be real.
This requires some care in the presence of multiple roots; but a complex root and its conjugate do have the same multiplicity. 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 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. It needs to be shown that
as well.
If P = 0, then
which can be put as
Now
and given the properties of complex conjugation,
Since
it follows that
That is,
Note that this works only because the ar are real, that is,. If any of the coefficients were non-real, the roots would not necessarily come in conjugate pairs. In addition, one can show that for any, it holds that even if.