This leads to the following theorem:

Given a polynomial $p(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0 $ of degree $n\geq 1$ where $a_i\in \mathbb{C}$ , there are exactly $n$ roots in $\mathbb{C}$ to the equation $p(x)=0$ if we count multiple roots.

Proof The non-constant polynomial $a_1x-a_0$ has one root, $x=a_0/a_1$ . Next, assume that a polynomial of degree $n-1$ has $n-1$ roots.

The polynomial of degree $n$ has then by the fundamental theorem of algebra a root $z_n$ . With polynomial division we find the unique polynomial $q(x)$ such that $p(x)=(x-z_n)q(x)$ . The original equation has then $1 + (n-1)=n $ roots. By induction, every non-constant polynomial of degree $n$ has exactly $n$ roots.

For example, $x^4=0$ has four roots, $x_1=x_2=x_3=x_4=0$ .