$p$ is polynomial, set bounded open set with at most $n$ components Assume $p$ is a non constant polynomial of degree $n$. Prove that the set $\{z:|(p(z))| \lt 1\}$ is a bounded open set with at-most $n$ connected components. Give example to show number of components can be less than $n$.
thanks.
EDIT:Thanks,I meant connected components.

For bounding the number of connected components, you can show that every connected component contains at least one root of your polynomial. To obtain that, you can apply the minimum modulus principle to the connected component, just like in a proof of the fundamental theorem of algebra.
