YEAR 9 · ALGEBRA · 15 MIN

Common-factor factorisation

Reverse expansion using a factor shared by every term.

In this lesson

Find the greatest common factor.
Divide each term by the extracted factor.
Check by expanding again.

Before you start

Factors of integers
Expanding brackets
Variable index laws

01 / LEARN

Understand the idea.

A sum becomes a product

Factorising reverses expansion: $3 x + 12 = 3 (x + 4)$ writes a sum as a product with the same value. The outside factor must be shared by every term; the bracket records what remains in each term.

To factorise fully by a common factor, choose the greatest common factor of the coefficients and the smallest power of each variable present in every term. No further common factor greater than 1 should remain in all bracket terms.

Check every term, including a remaining one

For $12 x^{2} + 18 x$ , the coefficients share $6$ and both terms contain at least one $x$ . Removing $6 x$ leaves $2 x$ and $3$ , producing $6 x (2 x + 3)$ . Expand your proposed product to check both terms and signs.

A factor taken out of an entire term leaves $1$ , not $0$ : $5 x + 5 = 5 (x + 1)$ . Factoring is an identity and does not make a variable nonzero; for example, $6 x (2 x + 3) = 12 x^{2} + 18 x$ remains valid at $x = 0$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Extract a numerical factor

Fully factorise $15 a - 25$ .

Worked solution

1
Identify the greatest common factor.
$15 a = 5 (3 a), - 25 = 5 (- 5)$
The coefficient greatest common factor is 5, and the constant term contains no a.
2
Write the shared factor outside a bracket.
$15 a - 25 = 5 (3 a - 5)$
The bracket contains the remaining factor from each original term.
3
Expand to check completeness.
$5 (3 a - 5) = 15 a - 25$
The original terms return, and 3a and -5 have no further common factor.

Answer

$5 (3 a - 5)$

EXAMPLE 02

Extract a variable power

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

Worked solution

1
Identify the common numerical and variable factors.
$8 p^{3} = 4 p^{2} (2 p), - 12 p^{2} = 4 p^{2} (- 3)$
The coefficient greatest common factor is 4 and the lowest shared power is p squared.
2
Extract the shared factor.
$8 p^{3} - 12 p^{2} = 4 p^{2} (2 p - 3)$
The remaining factors are 2p and -3; no further common factor remains.
3
Expand the product to verify.
$4 p^{2} (2 p - 3) = 8 p^{3} - 12 p^{2}$
The identity holds for every real p, including zero.

Answer

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

03 / YOUR TURN

Try it yourself.

Check your understanding

Fully factorise $9 x + 12$ .

Show answer & reasoning

$3 (3 x + 4)$

The greatest common factor of 9 and 12 is 3, and the constant term has no x factor. Therefore $9 x + 12 = 3 (3 x + 4)$ , with no further common factor inside.

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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
$12 \div 12 = 1$ .
$12 (a + 1) = 12 a + 12$ .
$12 (a + 1)$

Checking saved progress…

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 ↗

In this lesson

Understand the idea.

A sum becomes a product

Check every term, including a remaining one

See the method in action.

Extract a numerical factor

Identify the greatest common factor.

Write the shared factor outside a bracket.

Expand to check completeness.

Extract a variable power

Identify the common numerical and variable factors.

Extract the shared factor.

Expand the product to verify.

Try it yourself.

Check your understanding

Keep practising