Common-factor factorisation | Year 10

01 / LEARN

The idea, step by step

Factorising reverses expansion

To factorise a sum, write it as a product. Start by finding a factor shared by every term. Taking out the greatest common factor gives the most useful form. For variable terms, use only the powers that occur in every term.

Check every term

Divide each original term by the extracted factor to find the bracket entries. Include a constant $1$ when a whole term has been taken outside. A negative common factor can be useful when the leading term is negative. Expanding your answer should reproduce the original expression exactly.

A common trap

$6 x + 6 = 6 x (1 + 1)$ .

Instead: The constant $6$ has no $x$ factor. The common factor is $6$ , giving $6 (x + 1)$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

The greatest shared factor

Factorise $12 x^{2} y + 18 x y^{2}$ .

The greatest numerical factor is $6$ .
Each term includes one $x$ and one $y$ , so extract $6 x y$ .
Dividing the terms by $6 x y$ leaves $2 x$ and $3 y$ .

6 x y (2 x + 3 y)

Example 2

An outside negative

Factorise $- 8 p + 12$ .

A common factor is $- 4$ .
Divide: $- 8 p \div (- 4) = 2 p$ and $12 \div (- 4) = - 3$ .
Expand $- 4 (2 p - 3)$ to check.

- 4 (2 p - 3)

Check your understanding

Factorise $15 a^{2} - 10 a$ .

Show answer

$5 a (3 a - 2)$

Both terms contain $5 a$ ; the bracket is formed by division.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Factorise $8 x + 24$ .
2 marks
Factorise $14 a^{2} + 21 a$ .
2 marks
Factorise $18 x y - 12 x$ .
2 marks
Factorise $- 10 m^{2} + 15 m$ .
2 marks
A garden contains $4 x$ rows with $3 y + 2$ plants in each row. Its expanded plant count is $12 x y + 8 x$ . Factorise this count and identify the common factor.
3 marks
Sam writes $x^{3} + x^{2} = x^{2} (x)$ . Correct the answer and explain the missing term.
3 marks

Read the worked solutions

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

Factorise $8 x + 24$ .
2 marks
The greatest common factor is $8$ .
$8 (x + 3)$
Factorise $14 a^{2} + 21 a$ .
2 marks
Extract $7 a$ .
$7 a (2 a + 3)$
Factorise $18 x y - 12 x$ .
2 marks
Extract $6 x$ .
$6 x (3 y - 2)$
Factorise $- 10 m^{2} + 15 m$ .
2 marks
Extract $- 5 m$ ; $5 m (- 2 m + 3)$ is equivalent.
$- 5 m (2 m - 3)$
A garden contains $4 x$ rows with $3 y + 2$ plants in each row. Its expanded plant count is $12 x y + 8 x$ . Factorise this count and identify the common factor.
3 marks
Divide both terms by $4 x$ .
$4 x (3 y + 2)$ ; common factor $4 x$ .
Sam writes $x^{3} + x^{2} = x^{2} (x)$ . Correct the answer and explain the missing term.
3 marks
Dividing $x^{3}$ by $x^{2}$ gives $x$ .
Dividing $x^{2}$ by $x^{2}$ gives $1$ , which cannot be omitted.
$x^{2} (x + 1)$

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian Mathematics Curriculum

Year 10 algebra, including the approved fuller school-course treatment of rational expressions and formula restrictions; quadratic methods are in their own chapter.

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