# How to Factor Variable Expressions

Factorization means “rewriting an expression as a multiplication problem.” When you factorize an expression containing variables, you apply the rules of parentheses in reverse. This means that instead of multiplying something into parentheses, you create parentheses to place something on the outside.

Rule

### MovingaFactorOutsideParentheses

 $ab+ac=a\left(b+c\right)$

To understand how to factorize an expression with variables, you have to be in full control of the times tables, how to multiply numbers and variables, and the difference between a term and its factors.

Rule

### FactorizinganExpressionwithVariables

1.
Factorize all the terms to find common factors.
2.
Place the factors that are common for all the terms in front of a set of parentheses.
3.
Write the remaining factors of the terms inside the parentheses.

You can always check whether you factorized correctly by expanding the parentheses and comparing the result to what you started with.

Example 1

Factorize the expression $2x+4$

 $2x+4=2\cdot x+2\cdot 2=2\left(x+2\right)$

Example 2

Factorize the expression $3x-9$

 $3x-9=3\cdot x-3\cdot 3=3\left(x-3\right)$

Example 3

Factorize the expression $4{x}^{3}+8{x}^{2}-16x$

$\begin{array}{llll}\hfill & \phantom{=}4{x}^{3}+8{x}^{2}-16x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =4\cdot x\cdot x\cdot x+4\cdot 2\cdot x\cdot x-4\cdot 4\cdot x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =4x\left({x}^{2}+2x-4\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill 4{x}^{3}+8{x}^{2}-16x& =4\cdot x\cdot x\cdot x+4\cdot 2\cdot x\cdot x-4\cdot 4\cdot x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =4x\left({x}^{2}+2x-4\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 4

Factorize the expression $5ab-10b$

$\begin{array}{llll}\hfill 5ab-10b& =5b\cdot a-5b\cdot 2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =5b\left(a-2\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Remember, you can always place the sign outside the parentheses as well! Example 5 shows you how this is done.

Example 5

Factorize the expression $-{x}^{2}-x$

$\begin{array}{llll}\hfill -{x}^{2}-x& =-x\cdot x+-x\cdot 1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-x\left(x+1\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$