# 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.

$\begin{array}{l}0={x}^{2}\left(x-2\right)+9\left(x-2\right)\\ 0=\left(x-2\right)\left({x}^{2}+9\right)\\ 0=\left(x-2\right)\left(x+3i\right)\left(x-3i\right)\\ x=2\text{or}x=-3i\text{or}x=3i\end{array}$