Difference between revisions of "Complex conjugate root theorem"
Etmetalakret (talk | contribs) |
Etmetalakret (talk | contribs) |
||
Line 1: | Line 1: | ||
− | + | In [[algebra]], the '''complex conjugate root theorem''' states that if <math>P(x)</math> is a [[polynomial]] with [[real number | real coefficients]], then a [[complex number]] is a root of <math>P(x)</math> 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. | 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. |
Revision as of 14:16, 27 August 2021
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.