Complex Conjugate Root Theorem
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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 is to present 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 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 a root of
, as required.