Factorize

Just as we expanded expressions in the previous chapter, we sometimes need to perform the reverse process: factorization. Factorization is the process of writing an expression as a product of factors, often by introducing brackets.

Common Factor Laws

The common factor laws are the reverse of the distributive laws: instead of expanding brackets, we look for a common factor to "pull out" and introduce brackets.

Proposition Common Factor Law 1

$\textcolor{colorprop}{a}b+\textcolor{colorprop}{a}c=\textcolor{colorprop}{a}(b+c)\quad$ and $\quad\textcolor{colorprop}{a}b-\textcolor{colorprop}{a}c=\textcolor{colorprop}{a}(b-c)$

Example

Factorize $2x+2$.

Answer

$$\begin{aligned}2x+2&=\textcolor{colorprop}{2}\times x +\textcolor{colorprop}{2}\times 1\\ &=\textcolor{colorprop}{2}(x+1). \end{aligned}$$

Difference of Squares

Proposition Difference of squares

$\textcolor{colordef}{a}^2-\textcolor{colorprop}{b}^2 = (\textcolor{colordef}{a}-\textcolor{colorprop}{b})(\textcolor{colordef}{a}+\textcolor{colorprop}{b})$

Example

Factorize $x^2-9$.

Answer

$$\begin{aligned}x^2-9&= \textcolor{colordef}{x}^2-\textcolor{colorprop}{3}^2\\ & = (\textcolor{colordef}{x}-\textcolor{colorprop}{3})(\textcolor{colordef}{x}+\textcolor{colorprop}{3})\\ \end{aligned}$$

Method Square root

If a number is not squared, take its square root, then apply the formula.

Example

Factorize $x^2-3$.

Answer

$$\begin{aligned}x^2-3 & = \textcolor{colordef}{x}^2 - \left(\textcolor{colorprop}{\sqrt{3}}\right)^2 \\ & = \left(\textcolor{colordef}{x}-\textcolor{colorprop}{\sqrt{3}}\right)\left(\textcolor{colordef}{x}+\textcolor{colorprop}{\sqrt{3}}\right)\end{aligned}$$

Proposition Sum of Squares

A sum of squares $a^2+b^2$ cannot be factored.

Example

Factorize if possible: $x^2+1$.

Answer

Since $x^2+1=x^2+1^2$, this sum of squares cannot be factored.

Binomial Factorization

Proposition Binomial factorization

$$\begin{aligned}\textcolor{colordef}{a}^{2}+2 \textcolor{colordef}{a} \textcolor{colorprop}{b}+\textcolor{colorprop}{b}^{2} &= (\textcolor{colordef}{a}+\textcolor{colorprop}{b})^{2}\\ \textcolor{colordef}{a}^{2}-2 \textcolor{colordef}{a} \textcolor{colorprop}{b}+\textcolor{colorprop}{b}^{2} &= (\textcolor{colordef}{a}-\textcolor{colorprop}{b})^{2}\\ \end{aligned}$$

Example

Factorize $x^2+2x+1$.

Answer

$$\begin{aligned}x^2+2x+1 & = \textcolor{colordef}{x}^{2}+2 \times \textcolor{colordef}{x}\times \textcolor{colorprop}{1}+\textcolor{colorprop}{1}^{2}\\ &= (\textcolor{colordef}{x}+\textcolor{colorprop}{1})^{2}\\ \end{aligned}$$