# Factors

In general, a factor of a number $x$ is a number that can be multiplied by another number to get $x$ .   But, the definition changes a little according to what kind of math you're doing.

## For whole numbers $a$ and $n$ :

We say $a$ is a factor of $n$ if $ab=n$ for some whole number $b$ .

For example, $3$ is a factor of $21$ , since $3\cdot 7=21$ .

But $4$ is not a factor of $21$ , since there is no whole number $b$ for which $4b=21$ .

## For polynomials $p$ and $r$ :

We say $p$ is a factor of $r$ if $pq=r$ for some polynomial $r$ .

For example, $x+1$ is a factor of ${x}^{2}-2x-3$ , since

$\left(x+1\right)\left(x-3\right)={x}^{2}-2x-3$ .

But $x+2$ is not a factor of ${x}^{2}-2x-3$ , since there is no polynomial $q$ for which $\left(x+2\right)\left(q\right)={x}^{2}-2x-3$ .

## Common Factors

If two numbers (or polynomials) have a factor in common, then it is called a common factor.

For instance, the numbers $15$ and $33$ have $3$ as a common factor.

The polynomials

$4x+4$  and   ${x}^{2}-2x-3$

have $x+1$ as a common factor.

$4x+4=4\left(x+1\right)$

${x}^{2}-2x-3=\left(x+1\right)\left(x-3\right)$