The question is, how many times does $f(C_r)$ go around $0$ for $C_r = \{x \in \mathbb{C}: |x|=r\}$ and a polynomial with complex coefficients $f$.
The answer is clear to me when $f(x) = ax^n + b$ but I don't see how exactly middle terms affect the result. How to see this without a graph? Can this be found for any $r$?
Many thanks