The distributive law says that for any numbers A, B, and C, A\left(B+C\right)=AB+AC. 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 the following expression by taking out the highest common factor:42x-x^2

Worked Solution

Create a strategy

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

Apply the idea

The coefficients of the terms do not have any common factors other than 1, so we cannot factorise out any numbers. However, both terms have at least one x so we can factorise that out.

\displaystyle 42x-x^2	\displaystyle =	\displaystyle x\times 42 +x \times	Write the terms as products of x
	\displaystyle =	\displaystyle x\left(42-x\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

\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.

Example 3

Factorise the following expression: 2t^{2}k^{7}+18t^{9}k^{9}

Worked Solution

Create a strategy

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

Apply the idea

The coefficients have a common factor of 2, and both terms have factors of t^{2} and k^{7}. So we can factorise 2t^{2}k^{7} out.

\displaystyle 2t^{2}k^{7}+18t^{9}k^{9}	\displaystyle =	\displaystyle 2t^{2}k^{7} \times 1+ 2t^{2}k^{7}\left(18t^{9}k^{9} \right)	Write the terms as products of 2t^{2}k^{7}
	\displaystyle =	\displaystyle 2t^{2}k^{7}\left(1+9t^{7}k^{2}\right)	Factorise

Idea summary

We can use the distributive law to factorise an algebraic expression like AB+AC. This means means writing the expression with any common factors between the terms taken outside of the brackets: AB+AC=A(B+C) Factorising is the reverse of expanding.

Outcomes

ACMNA230

Factorise algebraic expressions by taking out a common algebraic factor

ACMNA231

Simplify algebraic products and quotients using index laws

ACMNA232

Apply the four operations to simple algebraic fractions with numerical denominators

ACMNA233

Expand binomial products and factorise monic quadratic expressions using a variety of strategies

1.05 Factorising

Ideas

Factorise algebraic expressions

Examples

Example 1

Example 2

Example 3

Outcomes

ACMNA230

ACMNA231

ACMNA232

ACMNA233

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