There are tons and tons of proofs of this fact. Probably the quickest I know is that the non-constant polynomial is a holomorphic map and by setting induces a holomorphic map between the Riemann sphere and itself . But, it is a common fact that a holomorphic map between Riemann surfaces where the domain is compact, must be surjective, and so is surjective. But, since we see that there must exist some such that . Thus, you have that has one zero, and the fact that it has comes from basic algebra.