Theorem.

Every non-constant complex Polynomial $x^n+a_{n-1} x^{n-1} + ... + a_1 x +a_0$ has a root/zero ($a_{n-1}, ..., a_0 \in \mathbb{C}$, $n \geq 1$).

For proof we use Lemma If a continuous map $f: S^1 \to X$ extends to a cont. map $F: B^2 \to X$ (that is $F|S^1 = f$). Then $f_*: \pi_1(S^1,1) \to \pi_1(X, f(1))$ is the trivial homomorphism.

See here