📐 Factorising

KS3 Mathematics Worksheet

🎓 Mini-Lesson: Factorising Algebraic Expressions

What is Factorising?

Factorising is the opposite of expanding brackets. Instead of multiplying out, we're putting expressions back into brackets by finding common factors. Think of it as "un-multiplying"!

Expanding vs Factorising:
Expanding: 3(x + 4) → 3x + 12
Factorising: 3x + 12 → 3(x + 4)
Finding the Highest Common Factor (HCF)

Step 1: Find what's common to all terms

Example 1: Factorise 6x + 12
  1. List the factors of each term:
    6x = 2 × 3 × x
    12 = 2 × 2 × 3
  2. Find the HCF: The common factors are 2 and 3, so HCF = 6
  3. Take the HCF outside the brackets: 6(? + ?)
  4. Divide each term by the HCF:
    6x ÷ 6 = x
    12 ÷ 6 = 2
  5. Write the answer: 6(x + 2)
✓ Check: Expand 6(x + 2) = 6x + 12 ✓
Example 2: Factorise 4x + 8y
  1. Find the HCF of 4x and 8y: HCF = 4
  2. Take 4 outside: 4(? + ?)
  3. Divide each term:
    4x ÷ 4 = x
    8y ÷ 4 = 2y
  4. Answer: 4(x + 2y)
Example 3: Factorise 3x² + 9x
  1. Find the HCF of 3x² and 9x:
    Both have 3 as a factor
    Both have x as a factor
    HCF = 3x
  2. Take 3x outside: 3x(? + ?)
  3. Divide each term:
    3x² ÷ 3x = x
    9x ÷ 3x = 3
  4. Answer: 3x(x + 3)
✓ Check: Expand 3x(x + 3) = 3x² + 9x ✓
Factorising with Subtraction
Example 4: Factorise 12x - 8
  1. Find the HCF of 12x and 8: HCF = 4
  2. Take 4 outside: 4(? - ?)
  3. Divide each term:
    12x ÷ 4 = 3x
    8 ÷ 4 = 2
  4. Answer: 4(3x - 2)
Important: Keep the minus sign inside the brackets!
🔑 Steps to Factorise:
  • Step 1: Find the HCF of all terms (numbers AND letters)
  • Step 2: Put the HCF outside brackets
  • Step 3: Divide each term by the HCF to find what goes inside
  • Step 4: Check by expanding your answer

💡 Remember: Always check your factorising by expanding the brackets - you should get back to where you started!

✏️ Practice Questions

Factorise each expression fully. Write your answer in the form a(b + c) or a(b - c)