Factorising
Factorising is the reverse of expanding brackets and is a key skill in GCSE Maths used to simplify expressions and solve equations. Factorising means writing an expression as a product of factors. When taking out a common factor, write it outside the brackets and divide each original term by it to find the terms inside.
Start by taking out common factors, then learn to factorise quadratics. The final quadratic method is a clearly labelled Higher extension.
What does factorising mean?
Factorising writes a sum or difference as a product. Taking out a common factor is one method; some quadratics factorise into two brackets.
Examples of factorised expressions:
When expanded, these give back the original expressions.
Example Question:
We will work through this step by step:
Factorise:
Exam Tip
Always look for the greatest common factor (GCF) — the largest number and variable that divides into every term.
Factorising fully is important for gaining full marks.
Method — Factorising by Taking Out a Common Factor
To factorise:
- Identify the highest common factor (HCF) of all terms
- Divide each term by that factor
- Write the common factor outside the brackets and the divided terms inside
Worked Example
Factorise:
Step 1: Find the common factor
Both terms are divisible by
Step 2: Divide each term by
Step 3: Write the factorised form
Worked Example 2 (With Variables)
Factorise:
Step 1: Common factor is
Step 2: Divide each term
Step 3: Write result
Worked Example 3 (Including Variable Factor)
Factorise:
Step 1: Common factors are and
Step 2: Divide each term by
Step 3: Write result
Worked Example 4 (With a Negative Term)
Factorise:
Step 1: Common factor is
Step 2: Divide each term
Step 3: Write result
Final answer:
The factorised expression is written as a product of factors and brackets.
Check your solution (important habit)
Expand your answer to ensure it matches the original expression.
Example:
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Factorising quadratics
A quadratic contains a squared term such as x² and no higher power of x. Expanding two linear brackets produces a quadratic; we now reverse that process. These first examples are for Foundation and Higher.
Worked example — A positive constant
Factorise x² + 7x + 12.
In (x + p)(x + q), the middle coefficient is p + q and the constant is pq. We need two numbers that add to 7 and multiply to 12.
The pair is 3 and 4. Therefore x² + 7x + 12 = (x + 3)(x + 4).
Check: x² + 4x + 3x + 12 = x² + 7x + 12.
Worked example — A negative constant
Factorise x² + 2x − 15. We need a product of −15 and a sum of 2.
One number must be positive and one negative. The pair is 5 and −3.
(x + 5)(x − 3).
Check: x² − 3x + 5x − 15 = x² + 2x − 15.
If the product is positive but the sum is negative, both numbers are negative: x² − 7x + 12 = (x − 3)(x − 4).
Worked example — Difference of two squares
Factorise x² − 25. Since 25 = 5², use (x − 5)(x + 5).
Expanding gives x² + 5x − 5x − 25. The middle terms cancel.
In general, a² − b² = (a − b)(a + b). This works for a difference, not a sum of squares.
Check for a common factor first: 2x² − 18 = 2(x² − 9) = 2(x − 3)(x + 3).
Higher extension — A coefficient before x²
Factorise 2x² + 7x + 3.
Multiply the first coefficient and constant: 2 × 3 = 6. Find two numbers with product 6 and sum 7: 6 and 1.
Split the middle term: 2x² + 6x + x + 3.
Group: 2x(x + 3) + 1(x + 3). Both groups contain x + 3.
(2x + 1)(x + 3).
Check by expanding: 2x² + 6x + x + 3 = 2x² + 7x + 3.
Factorising an expression does not ask you to find x. To solve an equation, you also need an equals sign and a value on the other side. Quadratic equations need their own solving methods.
Think Like a Mathematician
Factorising is the opposite of expanding.
Being able to move between these forms is essential for solving algebra problems efficiently.
Common mistakes to avoid
- Not taking out the greatest common factor
- Forgetting to divide every term
- Sign errors inside brackets
- Leaving the expression partially factorised
Always check by expanding.
Try one yourself
Factorise:
Think carefully before checking.
The highest common factor (HCF) of 9 and 6 is 3.
Divide each term by 3: 9x ÷ 3 = 3x and 6 ÷ 3 = 2.
Write 3 outside the bracket and the quotients inside.
Answer: 3(3x + 2).
Check by expanding: 3 × 3x + 3 × 2 = 9x + 6.
Exam Style Question
Factorise:
The coefficients 12 and 18 have HCF 6, and both terms contain x. The HCF of the terms is 6x.
Divide the terms by the common factor: 12x² ÷ 6x = 2x and 18x ÷ 6x = 3.
Write the common factor outside the bracket.
Answer: 6x(2x + 3).
Check by expanding: 6x × 2x + 6x × 3 = 12x² + 18x.
More practice — factorising fully
1. Factorise 15a − 20.
The HCF is 5. Divide both terms by 5: 15a ÷ 5 = 3a and −20 ÷ 5 = −4.
5(3a − 4). Expanding returns 15a − 20.
2. Factorise x² + 9x + 20.
We need product 20 and sum 9. The numbers are 4 and 5.
(x + 4)(x + 5). Expansion gives x² + 5x + 4x + 20.
3. Factorise x² − x − 12.
Product −12 and sum −1 gives −4 and 3.
(x − 4)(x + 3). Check: x² + 3x − 4x − 12.
4. Factorise x² − 49.
49 = 7², so use the difference of two squares.
(x − 7)(x + 7). The +7x and −7x terms cancel when expanded.
5. Higher: factorise 3x² + 10x + 3.
3 × 3 = 9. The pair with product 9 and sum 10 is 9 and 1.
3x² + 9x + x + 3 = 3x(x + 3) + (x + 3).
(3x + 1)(x + 3). Expanding verifies all three original terms.
Frequently Asked Questions
Why is factorising important?
Factorising makes expressions easier to solve, simplify, and analyse. It is especially important when solving quadratic equations and simplifying algebraic fractions.
What does factorising mean in algebra?
Factorising means writing an expression as a product. You may take out a common factor or find two brackets whose expansion gives the original expression.