fundamental theorem of algebra

Fundamental Theorem of Algebra

Let $a_k\in \mathbb{C}$ for $k=0,1,2,\dots n$, such that $a_n\ne 0$. Then the polynomial $P:\mathbb{C}\to \mathbb{C}$ defined by

$$\sum _{k=0}^n{a}_k{z}^k=a_n\prod _{k=1}^n(z-z_k)$$

has exactly $n$ roots $z_1,z_2,\dots ,z_n\in \mathbb{C}$, counted with multiplicity (i.e., some roots may be repeated).
In other words, every non-constant polynomial with complex coefficients has at least one complex root.