Common-factor factorisation | Year 9

01 / LEARN

The idea, step by step

A sum becomes a product

Since $3 (x + 4) = 3 x + 12$ , the reverse process gives $3 x + 12 = 3 (x + 4)$ . Factorising writes an expression as a product without changing its value.

For a fully factorised common-factor answer, take the greatest common factor of the coefficients and the lowest power of each variable present in every term. The bracket should not still contain a shared factor.

Check every term, including a remaining one

$12 x^{2} + 18 x$ has common factor $6 x$ . Division leaves $2 x$ and 3, so $12 x^{2} + 18 x = 6 x (2 x + 3)$ . Multiplying the bracket back out confirms the original expression.

If an entire term is taken out, it leaves 1, not 0. For instance, $5 x + 5 = 5 (x + 1)$ . Taking out a negative factor may make the first bracket term positive. These ideas involve a single common factor; quadratic factorisation belongs in its own topic.

A common trap

Writing zero after dividing a term by itself.

Instead: A term divided by itself is 1; retain that 1 in the bracket.

02 / WORKED EXAMPLES

See the thinking

Example 1

Extract a numerical factor

Fully factorise $15 a - 25$ .

The greatest common factor of 15 and 25 is 5.
Divide both terms by 5 to leave $3 a$ and $- 5$ .
The result is $5 (3 a - 5)$ .

5 (3 a - 5)

Example 2

Extract a variable power

Fully factorise $8 p^{3} - 12 p^{2}$ .

The shared coefficient factor is 4; the lowest shared power is $p^{2}$ .
Division by $4 p^{2}$ leaves $2 p - 3$ .
$4 p^{2} (2 p - 3)$ expands back to the original expression.

4 p^{2} (2 p - 3)

Check your understanding

Fully factorise $9 x + 12$ .

Show answer

$3 (3 x + 4)$

Take out 3 to leave $3 x + 4$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Factorise $4 x + 12$ .
2 marks
Fully factorise $18 a - 24$ .
2 marks
Fully factorise $7 p^{2} + 14 p$ .
2 marks
Fully factorise $15 x^{2} y - 10 x y^{2}$ .
3 marks
Factorise $- 6 x - 18$ using a negative factor.
2 marks
A student writes $12 a + 12 = 12 (a + 0)$ . Correct and check the result.
3 marks

Read the worked solutions

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

Factorise $4 x + 12$ .
2 marks
Divide both terms by 4.
$4 (x + 3)$
Fully factorise $18 a - 24$ .
2 marks
The greatest common coefficient factor is 6.
$6 (3 a - 4)$
Fully factorise $7 p^{2} + 14 p$ .
2 marks
Take out $7 p$ .
$7 p (p + 2)$
Fully factorise $15 x^{2} y - 10 x y^{2}$ .
3 marks
Both terms share $5 x y$ .
Dividing gives $3 x$ and $- 2 y$ .
$5 x y (3 x - 2 y)$
Factorise $- 6 x - 18$ using a negative factor.
2 marks
Divide both terms by $- 6$ .
$- 6 (x + 3)$
A student writes $12 a + 12 = 12 (a + 0)$ . Correct and check the result.
3 marks
$12 \div 12 = 1$ .
$12 (a + 1) = 12 a + 12$ .
$12 (a + 1)$

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Year 9 algebraic simplification and equivalent expressions, with consolidation of linear formula skills. Binomial expansion and quadratic factorisation/equations are in the separate Quadratics and Parabolas chapter.

Official curriculum source ↗

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