YEAR 10 · ALGEBRA · 15 MIN

Common-factor factorisation

Reverse the distributive law by identifying the greatest factor shared by every term.

In this lesson

Find numerical and variable common factors
Check a factorisation by expanding it

Before you start

Factors and multiples
Expanding brackets

01 / LEARN

Understand the idea.

Factorising reverses expansion

Factorising writes a sum as a product, reversing the distributive law. Find the greatest numerical factor of all coefficients, then include only variables shared by every term. For a shared variable, take its lowest power in those terms.

Check every term

The bracket entries tell you what must multiply the outside factor to recover each original term. Obtain them by dividing each term by that factor, keeping signs attached. If an entire term is extracted, its bracket entry is $1$ , not zero.

A negative outside factor can make the first bracket term positive. Expand the completed product as a check: recovering every original term confirms the factorisation, including any constant or negative term.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

The greatest shared factor

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

Worked solution

1
Find the greatest shared factor.
$\begin{aligned} 12 x^{2} y & = 6 x y (2 x) \\ 18 x y^{2} & = 6 x y (3 y) \end{aligned}$
The coefficients share $6$ , and both terms contain at least one $x$ and one $y$ .
2
Place the remaining factors in a bracket.
$12 x^{2} y + 18 x y^{2} = 6 x y (2 x + 3 y)$
The distributive law combines terms with the same outside factor.
3
Check by expanding.
$6 x y (2 x + 3 y) = 12 x^{2} y + 18 x y^{2}$
Both original terms and their powers are recovered.

Answer

$6 x y (2 x + 3 y)$

EXAMPLE 02

An outside negative

Factorise $- 8 p + 12$ .

Worked solution

1
Choose a negative common factor.
$- 8 p + 12 = (- 4) (2 p) + (- 4) (- 3)$
Both coefficients are divisible by $- 4$ ; the positive constant needs a negative bracket entry.
2
Extract that factor.
$- 8 p + 12 = - 4 (2 p - 3)$
The bracket records the two factors left after extracting $- 4$ .
3
Expand to check both signs.
$- 4 (2 p - 3) = - 8 p + 12$
The product of the two negatives reproduces the positive constant.

Answer

$- 4 (2 p - 3)$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$5 a (3 a - 2)$

The greatest shared factor is $5 a$ . The remaining factors are $3 a$ and $- 2$ , so $15 a^{2} - 10 a = 5 a (3 a - 2)$ . Expansion verifies the signs and powers.

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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.

Official curriculum source ↗

In this lesson

Understand the idea.

Factorising reverses expansion

Check every term

See the method in action.

The greatest shared factor

Find the greatest shared factor.

Place the remaining factors in a bracket.

Check by expanding.

An outside negative

Choose a negative common factor.

Extract that factor.

Expand to check both signs.

Try it yourself.

Check your understanding

Keep practising