The fundamental theorem of algebra pretty much states the following, every non-zero, single-variable, polynomial of degree \(n\) has exactly \(n\) roots even thought not all of them might be real (remember complex roots).

Proof:

Given a non-constant polynomial \(p(z)\) with complex coefficients, the polynomial \(q(z) = p(z)p(\bar{z})\) has only real coefficients and, if \(z\) is a zero of \(q(z)\), then either \(z\) or its conjugate is a root of \(p(z)\).

There is some positive real number \(\mathbb{R}\) such that:

Complex-analytic proof:

Topological proof:

Algebraic proof:

Geometric proof: