Please, help me!

Suppose that F is algebraically closed, p(z) is in F[z] and a is in F. Prove that a is an eigenvalue of p(T) if and only if a=p(lanbda) for some eigenvalue lambda of T. (Hint: Suppose that a is an eigenvalue of p(T). Factor p(z)-a=c(z-lambda_1)...(z-lambda_m). Use the fact that p(T)-aI is not injective. Don't forget to consider what happens if c=0).

Thank you in advance!