# Factorising

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 expression fully, means to put it in brackets by taking out the highest common factors.

The simplest way 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 there are different ways to factorise different types of algebraic expressions; we will learn about them all here.

### What is factorising? ## 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:

Remember: 3x+6 is known as a binomial because it is an expression with two terms

2. Factorising double brackets

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

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

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

3. Differences of two squares

Using the difference of two squares:

### Explain how to factorise expressions ## Factorisation methods

Each method of factorising or factoring 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 factorise fully:

$\color{#00BC89}3x + \color{#7C4DFF}6$

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

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.

$\color{#FF9100}3(\quad+\quad)$

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

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

$\color{#FF9100}3 \times \color{#62F030}x = \color{#00BC89}3 x$

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

$\color{#FF9100}3 \times \color{#92009E}2 = \color{#7C4DFF}6$

$\color{#FF9100}3(\color{#62F030}x + \color{#92009E}2)$

We can check the answer by multiplying out the bracket!

$3(x+2)=3 x+6$

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 factorise fully:

$x^2 + \color{#00bc89}6x + \color{#7C4DFF}5$

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.

$(x\qquad )(x\qquad)$

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.

$(x+\color{#FF9100}1)(x+\color{#FF9100}5)$

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

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

To factorise fully:

$\color{#FE47EC}2x^2 + \color{#00bc89}5x \color{#7C4DFF}{+3}$

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

Factors of 6:
1, 6
2, 3

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.

$2 x^{2}\color{#FF9100}{+2 x+3 x}+3$

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

$\color{#398CDA}{2 x}\color{#62F030}{(x+1)}\color{#398CDA}{+3}\color{#62F030}{(x+1)}$

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.

$(\color{#398CDA}{2x+ 3})\color{#62F030}{(x + 1)}$

Step by step guide: Factorising quadratics

### 3. Difference of two squares

Factorising example using difference of two squares:

To factorise fully:

$\color{#FE47EC}4x^2 -\color{#7C4DFF}{ 9}$

1. Write down 2 brackets.
$(\qquad)(\qquad)$

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

$\sqrt{\color{#FE47EC}{4 x^2}}=\color{#FE47EC}{2 x}$
$(\color{#FE47EC}{2x}\qquad)(\color{#FE47EC}{2x}\qquad)$

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

$\sqrt{\color{#7C4DFF}9}=\pm\color{#7C4DFF}3$
$(\color{#FE47EC}{2x}\quad\color{#7C4DFF}3)(\color{#FE47EC}{2x}\quad\color{#7C4DFF}3)$

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

$(\color{#FE47EC}{2x}+\color{#7C4DFF}3)(\color{#FE47EC}{2x}-\color{#7C4DFF}3)$

Step by step guide: Difference of two squares

## Practice factorising questions (mixed)

1. Fully factorise:

10-5y

5(2-y) 10(1-y) 5(2-5y) 5(2+y) The highest common factor of 10 and 5 is 5 . Therefore, we can divide the original expression by 5 , which means the bracket must contain 2-y .

2. Fully factorise:

20{x}^2-8x

4(5x^{2}-2x) 4x(5x-2) 2x(10x-4) 4x^{2}(5x-2) The highest common factor of 20x^{2} and 8x is 4x . Therefore, we can divide the original expression by 4x , which means the bracket must contain 5x-2 .

3. Fully factorise:

{x}^2-x-6

(x+2)(x-3) (x-2)(x-3) (x-2)(x+3) (x+1)(x-6) In this case, we are looking for numbers that multiply to make -6 and add to make -1 . By considering the factor pairs of -6 , we conclude that we can use +2 and -3 .

4. Fully factorise:

2{x}^2-4x-6

(2x+2)(2x-3) 2(x^{2}-2x-3) (2x+2)(x-3) 2(x+1)(x-3) The original expression can be divided by 2 , and then written as a product of two brackets.

2{x}^2-4x-6

Divide each term by 2 , and rewrite.

2(x^{2}-2x-3)

To factorise the quadratic expression, we are looking for numbers that multiply to -3 and sum to -2 . By considering factor pairs, we conclude that we need to use +1 and -3 .

2(x+1)(x-3)

5. Fully factorise:

{x}^2-9

(x+3)(x+3) (x+1)(x-9) (x+3)(x-3) (x+1)(x-3) This is a special case (difference of two squares), which means we can take square roots of the coefficient of x and the constant term, then write one bracket with a + sign and the other bracket with a sign.

6. Fully factorise:

9{x}^2-16

(3x+4)(3x-4) (3x-4)(3x-4) (9x+16)(x-1) (3x+1)(3x-16) This is a special case (difference of two squares), which means we can take square roots of the coefficient of x and the constant term, then write one bracket with a + sign and the other bracket with a sign.

## Factorising GCSE questions (mixed)

1. Factorise: 9x − 18

9(x − 2)

(1 mark)

2. Factorise fully: 16x2 + 20xy

4x(4x + 5y)

(2 marks)

3. Factorise fully: 3y2 − 4y − 4

(3y + 2)(y − 2)

(2 marks)

## 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)

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