Complex Conjugate Root Theorem
In algebra, the Complex Conjugate Root Theorem states that if is a polynomial with real coefficients, then a complex number is a root of if and only if its complex conjugate is also a root.
A common intermediate step in intermediate competitions is to recognize that when given a complex root of a real polynomial, its conjugate is also a root.
Proof
Let have the form for some real numbers and let be a complex root of . We wish to show that , the complex conjugate of , is also a root of . We have that Then by the properties of complex conjugation, which entails that is also a root of , as required.