fundamental theorem of algebra result

This leads to the following theorem:

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

ProofThe non-constant polynomial $a_{1}x-a_0$ 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, $x_1=x_2=x_3=x_4=0$.