Consider the polynomial of a complex variable with real coefficients.
- Prove that
for all
.
- Using part (a) show that if
has any nonreal zeros, they must occur in pairs of a complex number and its conjugate.
- Proof. Let
Then, using the properties of conjugation we have,
(The final line follows since
and then induction for to get
for all
.) This completes the proof
- Proof. If
is a non-real zero of
then
and
(since
is not real by assumption). Hence,
is also a zero of
, so the non-real zeros come in pairs