3 Types of Factorising

1. Factorise by Grouping

We have seen the distributive law: $A\left(B+C\right)=AB+AC$.

We have also seen how the distributive applies when the first term is a binomial expression: $\left(A+B\right)\left(C+D\right)=A\left(C+D\right)+B\left(C+D\right)$.

This means that if we have an expression such as $y\left(x+4\right)+5\left(x+4\right)$, we can write it as a product of two brackets: $\left(y+5\right)\left(x+4\right)$.

2. Factorise using Perfect Squares

We’ve seen how to expand perfect squares and use a model to visualise it:

$\left(A+B\right)^2=A^2+2AB+B^2$

Model:

When factorising an expression of the type $A^2+2AB+B^2$:

• Look for a common factor to both terms first as this can simplify the expression for the remaining steps.

• Identify values for A and B that satisfy this form.

• Write in factorised form using the rule $\left(A+B\right)^2=A^2+2AB+B^2$.

In our work in quadratics, the first term $A$, is often the variable $x$, so for the expression $x^2+4x+4$, we would identify $A=x$ and $B=2$, because $2^2=4$. The middle term $2AB$ holds true as $2\cdot x\cdot2=4x$.

3. Factorise using Difference of Two Squares

We know that the difference of two squares looks like this: $A^2-B^2=\left(A-B\right)\left(A+B\right)$.

When factorising an expression of the type $A^2-B^2$:

• Look for a common factor to both terms first as this can simplify the expression for the remaining steps.

• Identify the terms that are the squares $A$ and $B$.

• Write in factorised form using the rule $A^2-B^2=\left(A-B\right)\left(A+B\right)$.