Factorising common factors | Year 8

01 / LEARN

The idea, step by step

Reverse the distributive property

Factorising writes a sum as a product. In $8 x + 12$ , both terms contain $4$ . Dividing each term by $4$ leaves $2 x$ and $3$ , so $8 x + 12 = 4 (2 x + 3)$ .

The greatest common integer factor gives a fully factorised result for this type of expression. A smaller common factor may be correct but leave another common factor inside the bracket.

A variable can be common too

Both terms in $6 a^{2} + 9 a$ contain $3 a$ . Dividing the terms by $3 a$ gives $2 a$ and $3$ , so the expression is $3 a (2 a + 3)$ . Retain the signs of the quotients.

Expand your result to check that it recovers every original term. Factorising is an identity, not an instruction to solve an equation. A variable acquires no numerical value simply because a factor appears outside a bracket.

A common trap

Writing $10 x + 15 = 5 (2 x + 15)$ .

Instead: Every term must be divided by the extracted factor: $15 / 5 = 3$ , so write $5 (2 x + 3)$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Numerical common factor

Factorise fully $12 x + 18$ .

The greatest common factor is $6$ .
The quotients are $2 x$ and $3$ .
Check by expanding.

6 (2 x + 3)

Example 2

Algebraic common factor

Factorise fully $8 a^{2} - 12 a$ .

The terms share $4 a$ .
Dividing gives $2 a$ and $- 3$ .
Expansion recovers $8 a^{2} - 12 a$ .

4 a (2 a - 3)

Check your understanding

Factorise fully $6 x + 9$ .

Show answer

$3 (2 x + 3)$

Extract the greatest common factor $3$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Factorise fully $4 x + 8$ .
2 marks
Factorise fully $15 a - 10$ .
2 marks
Factorise fully $9 p + 12 q$ .
2 marks
Factorise fully $5 x^{2} + 20 x$ .
2 marks
Factorise fully $14 m^{2} - 21 m$ .
2 marks
A student writes $12 x + 18 = 3 (4 x + 6)$ . Is this correct and fully factorised?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Factorise fully $4 x + 8$ .
2 marks
Extract $4$ .
$4 (x + 2)$
Factorise fully $15 a - 10$ .
2 marks
Extract $5$ and preserve the subtraction.
$5 (3 a - 2)$
Factorise fully $9 p + 12 q$ .
2 marks
The coefficients share $3$ , but the variable parts differ.
$3 (3 p + 4 q)$
Factorise fully $5 x^{2} + 20 x$ .
2 marks
Both terms contain $5 x$ .
$5 x (x + 4)$
Factorise fully $14 m^{2} - 21 m$ .
2 marks
Both terms contain $7 m$ .
$7 m (2 m - 3)$
A student writes $12 x + 18 = 3 (4 x + 6)$ . Is this correct and fully factorised?
3 marks
Expansion gives the original expression.
The bracket still has common factor $2$ .
Correct, but not fully factorised: $6 (2 x + 3)$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 Mathematics with SA curriculum context

Combining like terms; substitution with signed/rational values; multiplying/dividing algebraic terms; expanding a single bracket; factorising common factors; equivalent linear expressions; rearranging simple formulas.

Official curriculum source ↗

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