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