Expanding & Factorising
Learn the distributive law, expand single and double brackets, and factorise by taking out common factors.
The Distributive Law
Expanding means removing brackets by multiplying each term inside the bracket by the term outside. The key rule is:
a(b + c) = ab + ac
Multiply a by each term inside the bracket.
Visual: Area Model for 3(x + 4)
Think of expanding as finding the total area of a rectangle:
Total area: 3x + 12
Expanding Double Brackets (FOIL)
To expand two brackets like (x + 2)(x + 3), multiply each term in the first bracket by each term in the second bracket. Use the FOIL method:
Combine: x² + 3x + 2x + 6
Collect like terms: x² + 5x + 6
Grid Method for (x + 2)(x + 3)
Factorising: Taking Out Common Factors
Factorising is the reverse of expanding. We look for a common factor shared by all terms and write it outside a bracket.
ab + ac = a(b + c)
Find the common factor (a) and divide each term by it.
Example: Factorise 6x + 9
Step 1: Find the highest common factor (HCF) of 6 and 9. HCF = 3
Step 2: Divide each term by 3: 6x ÷ 3 = 2x, and 9 ÷ 3 = 3
Result: 6x + 9 = 3(2x + 3)
Key Vocabulary
Expanding
Removing brackets by multiplying each term inside by the term outside.
Factorising
The reverse of expanding: writing an expression as a product of factors with brackets.
Distributive Law
The rule a(b + c) = ab + ac. Distribute the outside term to each inside term.
Common Factor
A number or variable that divides evenly into all terms of an expression.
Worked Examples
Expand: 4(2x − 5)
Step 1: Multiply 4 by the first term: 4 × 2x = 8x
Step 2: Multiply 4 by the second term: 4 × (−5) = −20
Answer: 8x − 20
Expand and simplify: (x + 4)(x − 2)
F: x × x = x²
O: x × (−2) = −2x
I: 4 × x = 4x
L: 4 × (−2) = −8
Combine: x² − 2x + 4x − 8
Answer: x² + 2x − 8
Factorise: 10x² + 15x
Step 1: Find HCF of 10x² and 15x. The HCF of the numbers is 5. Both terms contain x. So HCF = 5x
Step 2: Divide each term: 10x² ÷ 5x = 2x, and 15x ÷ 5x = 3
Answer: 5x(2x + 3)
Check: 5x × 2x = 10x² and 5x × 3 = 15x ✓
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
Expand: 5(3x + 2)
Question 2
Expand and simplify: (x + 5)(x + 1)
Question 3
Factorise: 8x + 12
Question 4
Expand: −2(3x − 4)
Question 5
Factorise fully: 12x² − 18x
Key Concepts Summary
- ● The distributive law: a(b + c) = ab + ac. Multiply the outside term by every term inside.
- ● For double brackets, use FOIL (First, Outer, Inner, Last) or a grid method, then collect like terms.
- ● Watch out for negative signs when expanding: negative × negative = positive.
- ● Factorising is the reverse of expanding: find the HCF and place it outside brackets.
- ● Always check your answer by expanding it back out to see if you get the original expression.