\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)

Factorization of Algebraic Expressions

Factorization is the reverse process of expansion. While expanding converts a product into a sum, factorization converts a sum into a product.
Mastering factorization is a crucial skill for simplifying complex expressions and solving higher-degree equations. Unless stated otherwise, we work with real numbers in this chapter.

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 Laws
\(\textcolor{colorprop}{a}b+\textcolor{colorprop}{a}c=\textcolor{colorprop}{a}(b+c)\quad\) and \(\quad\textcolor{colorprop}{a}b-\textcolor{colorprop}{a}c=\textcolor{colorprop}{a}(b-c)\)
Example
Factorize \(2x+2\).

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

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 \(x^2-9\).

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

Note
Any non-negative real number \(c\) can be written as a perfect square: \(c=(\sqrt{c})^2\). This allows us to rewrite expressions such as \(x^2 - c\) as a difference of squares and apply the formula above.
Example
Factorize \(x^2-3\).

We can write \(3\) as \((\sqrt{3})^2\) to create a difference of squares.$$\begin{aligned}x^2-3 &= \textcolor{colordef}{x}^2 - \left(\textcolor{colorprop}{\sqrt{3}}\right)^2 \\ &= \left(\textcolor{colordef}{x}-\textcolor{colorprop}{\sqrt{3}}\right)\left(\textcolor{colordef}{x}+\textcolor{colorprop}{\sqrt{3}}\right).\end{aligned}$$

Note
A sum of squares \(a^2+b^2\) cannot be factored over the real numbers.
Example
Factorize if possible: \(x^2+1\).

Since \(x^2+1=x^2+1^2\) is a sum of squares, it cannot be factored into linear factors with real coefficients.

Perfect Square Trinomials

Proposition Perfect Square Trinomials
$$\begin{aligned}\textcolor{colordef}{a}^{2}+2 \textcolor{colordef}{a} \textcolor{colorprop}{b}+\textcolor{colorprop}{b}^{2} &= (\textcolor{colordef}{a}+\textcolor{colorprop}{b})^{2}\\ \textcolor{colordef}{a}^{2}-2 \textcolor{colordef}{a} \textcolor{colorprop}{b}+\textcolor{colorprop}{b}^{2} &= (\textcolor{colordef}{a}-\textcolor{colorprop}{b})^{2}\\ \end{aligned}$$
Example
Factorize \(x^2+2x+1\).

We check if this fits the form \(a^2+2ab+b^2\).
  • The first term is \(x^2\), so let \(a=x\).
  • The last term is \(1=1^2\), so let \(b=1\).
  • We check if the middle term is \(2ab\): \(2(x)(1) = 2x\). It matches.
Therefore, the expression is a perfect square:$$\begin{aligned}x^2+2x+1 &= \textcolor{colordef}{x}^{2}+2(\textcolor{colordef}{x})(\textcolor{colorprop}{1})+\textcolor{colorprop}{1}^{2}\\ &= (\textcolor{colordef}{x}+\textcolor{colorprop}{1})^{2}.\end{aligned}$$