number of zeros of a complex polynomial's leading term

Question

Let $p(z)=a_n z^n + a_{n-1} z^{n+1}+...$ be a polynomial of degree $n$. Prove that in a disc of sufficiently large radius, $p(z)$ and $r(z)=a_n z^n$ have the same number of zeros.

Answer 1

The number of zeros of $p$ is finite, equal to the degree. Consider a disc of radius the largest modulus of a root, plus an epsilon.

Answer 2

hint:You can use Rouche's Theorem.