Exercise 4.6

Answers

Proof. Suppose $p$ has $m$ distinct zeros. Let $λ$ be a zero of $p$ . Then

p (z) = (z - λ) q (z)

For some polynomial $q$ . Using the product rule, we get

p^{'} (z) = q (z) + (z - λ) q^{'} (z)

Because the zeros of $p$ are distinct, it follows that $λ$ is not a zero of $q$ . Therefore

p^{'} (λ) = q (λ) + (λ - λ) q^{'} (λ) = q (λ) \neq 0

Thus $λ$ is not a zero of $p^{'}$ , as desired. □

celiopassos

2017-10-06 00:00