Prove that a polynomial in the complex plane of degree n can

Prove that a polynomial in the complex plane of degree n can have only n zeros.

Solution

(easily) proved by induction on n. Let f(z) be the given polynomial with degree n.

n=1. f(z) = az+b . so f(z)=0 has unique root -b/a. In other words, the number of roots =1.

Assume the statement for n.

Let f(z) have degree n+1.

If f(z) has no roots at all (but the fundamental theorem of algebra assures at least 1, because we are on the complex field) , there is nothing to prove.

Otherwise, let a be a root(zero). Then, by remainder theorem,

f(z) =(z-a)g(z) with degree g =n.

Applying induction, g(z) can have at most n zeros.

Hence f(z) can have at most n+1 zeros and we are done.

f(z