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: $ax + ay = a(x + y)$

infoNote

Example: $6x^2 + 9x = 3x(2x + 3)$

Step 1: Identify the common factor here, it's $\ 3x$ .
Step 2: Factor it out: $\ 6x^2 + 9x = 3x(2x + 3)$ .

2. Factorising Quadratic Expressions (Trinomials)

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

infoNote

Example 1: Factorising Simple Quadratics when $\ a = 1$ : $x^2 + 5x + 6$

Step 1: Identify two numbers that multiply to $\ 6$ (the constant term) and add to $\ 5$ (the coefficient of $\ x$ .
Step 2: The numbers are $\ 2$ and $\ 3$ (since $\ 2 \times 3 = 6$ and $\ 2 + 3 = 5 )$ .
Step 3: Write the factors: $\ (x + 2)(x + 3)$ .

infoNote

Example 2: Factorising Complex Quadratics when $a \neq 1$ : $2x^2 + 7x + 3$

Step 1: Multiply $\ a \times c$ (here, $\ 2 \times 3 = 6$ ).
Step 2: Find two numbers that multiply to $\ 6$ and add to $\ 7$ (they are $\ 6$ and $\ 1$ ).
Step 3: Rewrite the middle term: $\ 2x^2 + 6x + x + 3$ .
Step 4: Factor by grouping: $\ 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) .$

3. Difference of Squares

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

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

infoNote

Example: $x^2 - 9 = (x + 3)(x - 3)$

$\ x^2$ is a square, and $\ 9$ is a square ( $\ 9 = 3^2$ ).
Apply the difference of squares formula: $\ x^2 - 9 = (x + 3)(x - 3) .$

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

infoNote

Example: $x^2 + 6x + 9 = (x + 3)^2$

Recognise that $\ 6x$ is twice the product of $\ x$ and $\ 3$ , and $\ 9$ is $\ 3^2$ .
Factor as $\ (x + 3)^2 .$

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.

infoNote

Example: $ax + ay + bx + by$

Step 1: Group the terms: $\ (ax + ay) + (bx + by)$ .
Step 2: Factor each group: $\ a\ (x + y) + b\ (x + y)$ .
Step 3: Factor out the common binomial: $\ (a + b)(x + y)$ .

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.

Factorisation Simplified Revision Notes for A-Level OCR Maths Pure

2.5.4 Factorisation

1. Factorising by Taking Out the Common Factor

2. Factorising Quadratic Expressions (Trinomials)

3. Difference of Squares

4. Factorising Perfect Square Trinomials

5. Factorising by Grouping

Summary:

500K+ Students Use These Powerful Tools to Master Factorisation For their A-Level Exams.

Other Revision Notes related to Factorisation you should explore