# Fundamental Theorem of Algebra

There are a couple of ways to state the Fundamental Theorem of Algebra. One way is:

A polynomial function with complex numbers for coefficients has at least one zero in the set of complex numbers .

A different version states:

An $n$ th degree polynomial function with complex coefficients has exactly $n$ zeros in the set of complex numbers, counting repeated zeros .

Note: The real numbers are a subset of the complex numbers, since every real number can be written in $a+bi$ form, with $b=0$ . So, the theorem is also true for polynomials with real coefficients.

Example:

$g\left(x\right)={x}^{3}-2{x}^{2}+9x-18$

In this case, the coefficients are all real numbers: $3,-2$ and $9$ .

Set $g\left(x\right)=0$ and factor over the complex numbers to find the zeros.