Deciding the Factorization Method
GCSE Mathematics🧩 How to Choose the Right Method
Not all expressions factorize the same way. The method you choose depends on the number of terms, the signs, and the coefficients. This page will help you decide which factorization method to use for any given expression.
Key Questions to Ask:
1. How many terms?
2. Is there a common factor?
3. Is it a difference of squares?
4. Is it a quadratic?
5. Is the coefficient of x² greater than 1?
1. How many terms?
2. Is there a common factor?
3. Is it a difference of squares?
4. Is it a quadratic?
5. Is the coefficient of x² greater than 1?
🌳 Factorization Decision Tree
Start: Algebraic Expression
Step 1: Common factor?
Check if all terms share a common factor
↓
Step 2: How many terms?
2 terms, 3 terms, or 4+ terms
2 terms?
3 terms?
4+ terms?
2 Terms
Check for difference of squares
3 Terms
Quadratic: x² + bx + c or ax² + bx + c
4+ Terms
Try grouping in pairs
📋 Summary of Factorization Methods
1️⃣ Common Factor
When to use: All terms share a common factor
6x + 9 = 3(2x + 3)
4x² - 8x = 4x(x - 2)
2️⃣ Difference of Squares
When to use: Two terms, minus sign, both perfect squares
x² - 9 = (x - 3)(x + 3)
4x² - 25 = (2x - 5)(2x + 5)
3️⃣ Simple Quadratics (a = 1)
When to use: x² + bx + c form
x² + 7x + 12 = (x + 3)(x + 4)
x² - 5x + 6 = (x - 2)(x - 3)
4️⃣ Harder Quadratics (a > 1)
When to use: ax² + bx + c with a > 1
2x² + 7x + 3 = (x + 3)(2x + 1)
3x² + 10x + 8 = (x + 2)(3x + 4)
5️⃣ Factor by Grouping
When to use: Four or more terms
x³ + 3x² + 2x + 6 = (x + 3)(x² + 2)
x³ - 2x² + 3x - 6 = (x - 2)(x² + 3)
6️⃣ Multiple Methods
When to use: Need to apply more than one method
2x² - 18 = 2(x² - 9) = 2(x - 3)(x + 3)
4x³ - 36x = 4x(x² - 9) = 4x(x - 3)(x + 3)
🤔 Interactive Method Decider
Enter an expression and we'll help you choose the right method!
📊 Quick Decision Flowchart
Start
↓
Common factor?
YES
→
Factor it out first
NO
→
How many terms?
2 terms
Check: Difference of squares?
YES → (a-b)(a+b)
NO → Cannot factorize
3 terms
Quadratic
a = 1?
YES → Find factors of c
NO → Split middle term
4+ terms
Try grouping
Group in pairs
Look for common bracket
🎯 Practice: Choose the Correct Method
For the expression: x² + 7x + 12
Which factorization method should you use?
Common Factor
Difference of Squares
Simple Quadratic (a=1)
Harder Quadratic (a>1)
Factor by Grouping
Multiple Methods
Correct choices: 0/0
📋 Method Comparison Table
| Method | When to Use | Example |
|---|---|---|
| Common Factor | All terms share a factor | 6x + 9 = 3(2x + 3) |
| Difference of Squares | Two terms, minus, perfect squares | x² - 9 = (x-3)(x+3) |
| Simple Quadratic | x² + bx + c (a=1) | x² + 7x + 12 = (x+3)(x+4) |
| Harder Quadratic | ax² + bx + c (a>1) | 2x² + 7x + 3 = (x+3)(2x+1) |
| Grouping | Four or more terms | x³ + 3x² + 2x + 6 = (x+3)(x²+2) |
| Multiple Methods | Need to apply >1 method | 2x² - 18 = 2(x-3)(x+3) |