If a polynomial has real coefficients, and some of the roots are conplex, then the complex conjugate of each root is also a root.
To illustrate this for quadratis, notice that the roots of
are
and
Since
and
are complex.
and
Obviously
and
This observation can be generalised to polynomials of any order with real coefficients. If
is a root of a polynomial then so is
Proof: Let
(1) with
real.
Letbe a root, then
(1)
Taking the complex conjugate gives
(2) (ince all the coefficients are real).
Obviously
is then also a root.
Conversely supposeandare distinct roots.
(3) and
(4)
Taking the complex conjugate of the first one gives
(5)
(5) – (4) gives
Taking the complex conjugate gives
Adding gives
(6)
Divide by
and add to the complex conjugate. Do this repeatedly until you end up with
Repeat this process for decreasing powers of
to obtain
for all