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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:

Factorising

3x + 6 = 3(x + 2)

Factorising

2. Factorising double brackets

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

Factorising

x2 + 6x + 5 = (x + 5)(x + 1)

Factorising

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

Factorising

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

Factorising

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

Factorising

4x2 – 16 = (2x – 4)(2x + 4)

Factorising

What are the 3 standard ways to factorise algebraic expressions?

3 ways to factorise algebraic expressions:

  1. Factorising to single brackets
  2. Factorising quadratics into double brackets
  3. Factorising using the difference of two squares

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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.

Factors of 3:
1, 3

Factors of 6:
1, 6
2, 3


Top tip:

Writing the factor pairs makes it easier to list all the factors

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:

4x29

  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

Practice factorising questions (mixed)

1. Fully factorise: 10 – 5y

Show answer

=5(2 – y)

2. Fully factorise: 20x2 – 8x

Show answer

=4x(5x -2)

3. Fully factorise: x2 – x – 6

Show answer

=(x + 2)(x – 3)

4. Fully factorise: 2x2 – 4x – 6

Show answer

=(2x + 2)(x – 3)

5. Fully factorise: x2 – 9

Show answer

=(x + 3)(x-3)

6. Fully factorise: 4x2 – 16

Show answer

=(2x + 4)(2x – 4)

Factorising GCSE questions (mixed)

1. Factorise 9x – 18

Show answer

9(x – 2)

(1 mark)

2. Factorise fully 16x2 + 20xy

Show answer

4x(4x + 5y)

(2 marks)

3. Factorise fully 3y2 – 4y – 4

Show answer

(3y + 2)(y – 2)

(2 marks)

Factorising worksheets

Get your free factorising worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Learning Checklist

You have now learnt how to:

  • Manipulate algebraic expressions by taking out common factors to factorise into a single bracket.
  • Factorise quadratic expressions of the form x2 + bx + c
  • Factorise quadratic expressions of the form of the difference of two squares.
  • Factorise quadratic expressions of the form ax2 + bx + c (H)

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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