Show that $f(z)=\alpha$ has exactly $n$-distinct roots except for finitely many $\alpha \in \mathbb C$. 
Let $f(z)$ be a complex polynomial then show that except for finitely many $\alpha \in \mathbb C$ the equation $f(z)=\alpha$ has $n$-distinct roots except for finitely many $\alpha \in \mathbb C$, where $n$ is the degree of $f$.

I have tried my level best but I have failed to do it. How can I show that? Please help me. Thank you in advance.

$z$ is a multiple root is equivalent to $P(z)-\alpha=0$ and $P'(z)=0$, there are finitely many $z$ such that $P'(z)=0$, $c_1,...,c_p$, which are the potential multiple roots of $P(z)-\alpha$.
