Factorisation
Common Factors & Regrouping
To factorise, take out the highest common factor first: 6x + 9 = 3(2x + 3). When terms have no single common factor, group them: ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y).
Example 1: Factorise 6x + 9.
3(2x + 3).
Example 2: Factorise ax + ay + bx + by.
(a + b)(x + y).
Quick recap
- Take out the highest common factor first.
- Use regrouping when there is no single common factor.
✓ Quick check
Factorise 4x + 8.
The HCF is 4: 4(x + 2).
Factorise x² + x.
x is common: x(x + 1).
Factorisation Using Identities
Identities help factorise: a² − b² = (a + b)(a − b) and a² + 2ab + b² = (a + b)². So x² − 9 = (x + 3)(x − 3) and x² + 6x + 9 = (x + 3)².
Example 1: Factorise x² − 9.
(x + 3)(x − 3).
Example 2: Factorise x² + 6x + 9.
(x + 3)².
Quick recap
- a² − b² = (a + b)(a − b).
- a² + 2ab + b² = (a + b)².
✓ Quick check
Factorise x² − 25.
Difference of squares: (x + 5)(x − 5).
Factorise x² − 16.
Difference of squares: (x + 4)(x − 4).
Quadratic Trinomials & Division
To factorise a trinomial like x² + 5x + 6, split the middle term into two numbers that multiply to 6 and add to 5 (2 and 3): x² + 2x + 3x + 6 = (x + 2)(x + 3).
To divide algebraic terms, divide the coefficients and subtract the exponents: 6x² ÷ 2x = 3x.
Example 1: Factorise x² + 7x + 12.
Numbers 3 and 4: (x + 3)(x + 4).
Example 2: Divide 6x² by 2x.
3x.
Quick recap
- Split the middle term: two numbers with the right product and sum.
- Divide: divide coefficients, subtract exponents.
✓ Quick check
Factorise x² + 5x + 6.
2 × 3 = 6 and 2 + 3 = 5.
Divide 12x³ by 3x.
12 ÷ 3 = 4 and x³ ÷ x = x², so 4x².
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