Complex Conjugate Root Theorem
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.
Let have the form , where constants are real numbers, and let be a complex root of . We then wish to show that , the complex conjugate of , is also a root of . Because is a root of , Then by the properties of complex conjugation, which entails that is also a root of , as required.