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8.09 Factorising

Lesson

Factorisation

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 8Write 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 5Write 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.

Outcomes

MA4-8NA

generalises number properties to operate with algebraic expressions

MA4-9NA

operates with positive-integer and zero indices of numerical bases

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