Factorising common factors | Year 8

01 / LEARN

Understand the idea.

Reverse the distributive property

Factorising reverses expansion by writing a sum as a product. Find a factor shared by every term, place it outside a bracket, and work out what it multiplies to produce each original term. For example, $8 x + 12 = 4 (2 x + 3)$ .

For these expressions, factorise fully by taking out the greatest common numerical factor and all shared variable factors. A smaller extracted factor may give an equivalent expression but leave another common factor inside the bracket.

A variable can be common too

A variable can also be common. The terms $6 a^{2}$ and $9 a$ share $3 a$ , leaving $2 a$ and $3$ inside the bracket. Thus $6 a^{2} + 9 a = 3 a (2 a + 3)$ . Check signed terms carefully when the original sum includes subtraction.

Expand the proposed factorisation to recover every original term. This verifies an identity for all inputs, including zero; extracting a variable factor does not restrict that variable. There is no equation to solve unless an equality with another expression is actually given.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Numerical common factor

Factorise fully $12 x + 18$ .

Worked solution

1
Identify the greatest shared numerical factor.
$12 = 6 \times 2, 18 = 6 \times 3$
Six is the greatest common factor of the two coefficients, and the constant has no x factor.
2
Write each term using that factor.
$12 x + 18 = 6 (2 x) + 6 (3)$
Each original term must contain the same extracted factor.
3
Place the common factor outside one bracket.
$12 x + 18 = 6 (2 x + 3)$
Reversing distribution leaves no further common integer factor inside.
4
Expand to check.
$6 (2 x + 3) = 12 x + 18$
Both original terms are recovered exactly.

Answer

$6 (2 x + 3)$

EXAMPLE 02

Algebraic common factor

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

Worked solution

1
Find the common numerical and variable factors.
$8 a^{2} = (4 a) (2 a), - 12 a = (4 a) (- 3)$
The greatest coefficient factor is four, and both terms contain at least one a.
2
Extract the shared factor.
$8 a^{2} - 12 a = 4 a (2 a - 3)$
The remaining factors are two-a and negative three, preserving the subtraction.
3
Check by expanding the product.
$\begin{aligned} 4 a (2 a - 3) & = (4 a) (2 a) + (4 a) (- 3) \\ = 8 a^{2} - 12 a \end{aligned}$
The bracket terms share no further common factor, and the identity also holds at a equal to zero.

Answer

$4 a (2 a - 3)$

03 / YOUR TURN

Try it yourself.

Check your understanding

Factorise fully $6 x + 9$ .

Show answer & reasoning

$3 (2 x + 3)$

The greatest common factor of $6$ and $9$ is $3$ . Since $6 x = 3 (2 x)$ and $9 = 3 (3)$ , the fully factorised expression is $3 (2 x + 3)$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Reverse the distributive property

A variable can be common too

See the method in action.

Numerical common factor

Identify the greatest shared numerical factor.

Write each term using that factor.

Place the common factor outside one bracket.

Expand to check.

Algebraic common factor

Find the common numerical and variable factors.

Extract the shared factor.

Check by expanding the product.

Try it yourself.

Check your understanding

Keep practising