Complex conjugate root theorem
Revision as of 19:48, 21 August 2021 by Etmetalakret (talk | contribs)
The complex conjugate root theorem states that if is a polynomial with real coefficents, then a complex number is a root of
if and only if its complex conjugate is also a root.
A common setup in contest math is presenting a complex root of a real polynomial without its conjugate. It is then up to the solver to recognize that its conjugate is also a root.
Proof
Let have the form
, where
are real numbers. 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 a root of
, as required.