The distributive law says that for any numbers A,B, and C, A\left(B+C\right)=AB+AC. We saw how to use this rule to expand algebraic terms , but we can also use the rule in reverse.

The reverse of expanding algebraic expressions is called factorising. Factorising an algebraic expression means writing the expression with any common factors between the terms taken outside of the brackets.

Examples

Example 1

Factorise 45t-40.

Worked Solution

Create a strategy

Factorise using the law, AB-AC=A\left(B-C\right).

Apply the idea

5 is a factor of both 45 and 40, so we can factorise 5 out.

\displaystyle 45t-40	\displaystyle =	\displaystyle 5\times 9t - 5\times 8	Write the terms as products of 5
	\displaystyle =	\displaystyle 5 \left(9t-8\right)	Factorise

Example 2

Factorise the expression -2s-10.

Worked Solution

Create a strategy

Factorise using the law, AB+AC=A\left(B+C\right).

Apply the idea

-2 is a factor of both -2 and -10, so we can factorise -2 out.

\displaystyle -2s-10	\displaystyle =	\displaystyle -2\times s+\left(-2\right)\times 5	Write the terms as products of -2
	\displaystyle =	\displaystyle -2\left(s+5\right)	Factorise

Reflect and check

It is important to be careful about factorising a negative because the sign of each term will change. If the leading term in an expression is negative, then we should factorise the negative out.

Idea summary

We can use the reverse of distributive law to factorise an algebraic expression like so.AB+AC=A\left(B+C\right)

This means writing the expression with any common factors between the terms taken outside of the brackets. Factorising is the reverse of expanding.

4.05 Factorising

Ideas

Factorisation

Examples

Example 1

Example 2

Outcomes

MA4-8NA

MA4-9NA

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