 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.