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.