Factoring using notable identities

Reading identities backwards

Factoring means transforming a sum into a product. The notable identities can also be read from right to left:

Expanded form	Factored form
a² + 2ab + b²	(a+b)²
a² - 2ab + b²	(a-b)²
a² - b²	(a+b)(a-b)

Example: recognising a square

Let’s factorise x² + 6x + 9. We recognise that a² = x², so a = x; b² = 9, so b = 3; and 2ab = 2 × x × 3 = 6x, which corresponds to the middle term. Therefore: x^2 + 6x + 9 = (x+3)^2

Example: difference of two squares

Let’s factorise 4x^2 - 25. We write 4x² = (2x)² and 25 = 5², so a = 2x and b = 5: 4x² - 25 = (2x + 5)(2x - 5)

3-step method

1. Check that the two outer terms are squares (a² and b²)
2. Check the sign and value of the middle term (2ab), if present
3. Write the corresponding factored form directly

Pitfall to avoid

x² + 9 CANNOT be factored using the notable identities: the term 6x is missing, which should accompany a perfect square of the form (x+3)², and a sum of two squares can never be factored in the real numbers. Always check all three conditions before concluding that a factorisation is possible.