Common Factor

Find the highest common factor of all terms and put it outside a bracket.

Formula: ab + ac = a(b + c)

Example: 6x + 12 = 6(x + 2) 4x² + 8x = 4x(x + 2)

Fully Factorise

Examiners want the HIGHEST common factor. 2(2x+4) is factorised, but 4(x+2) is FULLY factorised.

Checking with Expansion

You can check your factorisation by multiplying the brackets out again.

Letters as Factors

Sometimes the common factor is a letter or both a number and a letter.

Example: 5x² - 10x = 5x(x - 2)

Factorising Quadratics

Find two numbers that multiply to give c and add to give b in x² + bx + c.

Formula: x² + bx + c = (x + p)(x + q) where p × q = c and p + q = b

Example: x² + 7x + 12 Numbers: 3 and 4 (3×4=12, 3+4=7) = (x + 3)(x + 4)

Difference of Two Squares

If the expression is a² − b², it factorises to (a+b)(a−b).

Formula: a² − b² = (a + b)(a − b)

Example: x² − 25 = (x + 5)(x − 5) 4x² − 9 = (2x + 3)(2x − 3)

Quadratics with a > 1

For ax² + bx + c where a > 1, use the "split the middle" method or trial and improvement.

Example: 2x² + 5x + 3 = (2x + 3)(x + 1)