How do I prove this using the hint?
THEOREM 97 (The fundamental theorem of algebra). Suppose p is a polynomial of degree n - 1. Then there exist numbers a and z1, z2, ..., zn (not necessarily distinct) such that p(z) = a(z - z1)(z - z2)...(z - zn).
HINT: One proof uses the fact that 1/p would be entire if p had no zeros and studies its behavior near infinity. Another proof uses Rouche's theorem.