Factorising – GCSE Maths

Here is everything you need to know about factorising for GCSE maths (Edexcel, AQA and OCR). You’ll learn the essentials of factorising expressions and factorising quadratics including factorising into single brackets and double brackets.

Look out for the factorising worksheets and exam questions at the end

What is factorising

Factorising is the reverse process of expanding brackets. To factorise an algebraic expression means to put it in brackets by taking out the common factors.

The simplest form of factorising is

• Find the highest common factor of each of the terms in the expression.
• Write the highest common factor (HCF) in front of any brackets
• Fill in each term in the brackets by multiplying out.

However you will need to learn various ways of factorising expressions depending on the examples you’re presented with.

How to factorise expressions

To factorise algebraic expressions there are three basic methods. When you are factorising quadratics you will usually use the double brackets or difference of two squares method.

1. Factorising single brackets

Example of factorising an algebraic expression:

3x + 6 = 3(x + 2)

2. Factorising double brackets

a) Example of factorising quadratic expressions in the form x² + bx + c

x² + 6x + 5 = (x + 5)(x + 1)

Remember:
Expressions with three terms like x²+ 6x + 5 and 2x²+ 5x + 3 are known as a trinomials.

b) Example of factorising quadratic expressions in the form ax² + bx + c

2x² + 5x + 3 = (2x + 3)(x + 1)

3. Factorising differences of two squares
Example of factorising the difference of two squares:

4x² – 16 = (2x – 4)(2x + 4)

Factorisation methods

Each method of factorising expressions is summarised below. For detailed examples, practice questions and worksheets on each one follow the links to the step by step guides.

1. Factorising single brackets

Factorising example using single brackets

To fully factorise:

3x + 6

1. Find the highest common factor (HCF) of the numbers 3 and 6.

The highest common factor (HCF) of 3x and 6 is 3

2 Write the highest common factor (HCF) at the front of the single bracket.

3( + )

3 Fill in each term in the bracket by multiplying out.

What do I need to multiply 3 by to give me 3x? x

What do I need to multiply 3 by to give me 6? 2

3(x + 2)

We can check the answer by multiplying out the bracket!

3(x + 2) = 3x + 6

Step by step guide: Factorising single brackets

2a) Factorising quadratics into double brackets: x² + bx + c

Factorising example for quadratic expressions in the form x² + bx + c

To fully factorise:

x² + 6x + 5

1. Write out the factor pairs of the last number (5)

Factors of 5:
1, 5

2 Find a pair of factors that + to give the middle number (6) and ✕ to give the last number (5).

1 + 5 = 6

1 ✕ 5 = 5

3 Write two brackets and put the variable at the start of each one.

(x )(x )

4 Write one factor in the first bracket and the other factor in the second bracket. The order isn’t important, the signs of the factors are.

(x + 1)(x + 5)

2b) Factorising quadratics with double brackets: ax² + bx + c

Factorising example for quadratic expressions in the form ax² + bx + c

To fully factorise:

2x² + 5x + 3

1. Multiply the the end numbers together (2 and 3) then write out the factor pairs of this new number in order

1 ✕ 6 = 6

2 ✕ 3 = 6

2 We need a pair of factors that + to give the middle number (5) and ✕ to give this new number (6)

2 + 3 = 5 ✔

2 ✕ 3 = 6 ✔

3 Rewrite the original expression, this time splitting the middle term into the two factors we found in step 2.

2x² + 2x + 3x + 3

4 Split the equation down the middle and fully factorise each half.

2x(x + 1) + 3(x + 1)

5 Factorise the whole expression by bringing whatever is in the bracket to the front and writing the two other terms in the other bracket.

(2x + 3)(x + 1)

Step by step guide: Factorising quadratics

3. Difference of two squares

Factorising example using difference of two squares:

To fully factorise:

4x² – 9

1. Write down 2 brackets.

( )( )

2 Square root the first term and write it on the left-hand side of both brackets.

√4x² = 2x

(2x )(2x )

3 Square root the last term and write it on the right-hand side of both brackets.

√9 = 3

(2x 3)(2x 3)

4 Put + in the middle of one bracket and – in the middle of the other (the order doesn’t matter).

(2x + 3)(2x – 3)

Step by step guide: Difference of two squares

The next topics are

• Solving quadratic equations by factoring
• Solving quadratics equations using the formula
• Forming and solving quadratic equations
• Completing the square

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