Factorisation (AQA A-Level Mathematics): Revision Notes

2.5.4 Factorisation

Factorisation is the reverse process of expanding brackets. It involves expressing an algebraic expression as a product of its factors, which simplifies expressions and solves equations. Understanding how to factorise correctly is essential for solving quadratic equations, simplifying fractions, and working with algebraic expressions.

1. Factorising by Taking Out the Common Factor

When all terms in an expression share a common factor, you can factorise by taking this factor outside the bracket.

General Form: $axe + ay = a(x + y)$

2. Factorising Quadratic Expressions (Trinomials)

Quadratic expressions are of the form $\ ax^2 + bx + c .$ The goal is to factorise it into two binomials.

3. Difference of Squares

A difference of squares is a special form that can be factorised into two binomials.

General Form: $a^2 - b^2 = (a + b)(a - b)$

4. Factorising Perfect Square Trinomials

A perfect square trinomial is of the form$\ a^2 + 2ab + b^2$, and it factorises into a squared binomial.

General Form: $a^2 + 2ab + b^2 = (a + b)^2$

5. Factorising by Grouping

This method is useful when an expression has four terms. Group the terms in pairs, factor each pair, and then factor the common binomial factor.

Summary:

• Common Factor: Factor out the greatest common factor.
• Quadratic Trinomials: Factor into binomials using middle term splitting.
• Difference of Squares: Apply the formula $\ a^2 - b^2 = (a + b)(a - b) .$
• Perfect Square Trinomials: Recognise and factor as$\ (a + b)^2$.
• Grouping: Group terms to factor by common factors.