 # Factorise Into a Single Bracket

In this worksheet, students will factorise expressions (i.e. add brackets in) by finding highest common factors in terms of numbers. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Notation, Vocabulary and Manipulation, Algebraic Expressions

Difficulty level:   ### QUESTION 1 of 10

Factorising is the opposite of expanding brackets.

To factorise an expression, you need to look for a common factor.

It might be a number or a letter.

Let's look at an example to see this in action.

e.g. Factorise 12x - 18y.

Look at the numbers.

We need to find the highest common factor (HCF) of 12 and 18.

The factors of 12 are: 1, 2, 3, 4, 6, 12, etc...

The factors of 18 are: 1, 2, 3, 6, 9, 18, etc...

The highest factor which they have in common is 6.

Place the number '6' goes outside of the bracket like this:

6( __  _  __ )

Now we need to work out what goes inside of the bracket.

Remember we are aiming to writing an expression with exactly the same value as 12x - 18y.

6 × 2x = 12x, so we need to put 2x in the bracket.

6 × -3y = -18y, so we need to put -3y in the bracket.

12x - 18y = 6(2x - 3y)

In this activity, we will factorise expressions by finding highest common factors in terms of numbers.

What is the highest common factor (HCF) of 24 and 18?

1

2

4

6

Match each expression below to its correct, factorised form.

## Column B

5a + 15b
5(5a - b)
25a - 5b
5(4a - b)
15a - 10b
5(3a - 2b)
15a - 20b
5(a + 3b)
20a - 5b
5(3a - 4b)

Factorise:

4a - 12b

Is it possible to factorise 8y + 3x?

Yes

No

Cannot tell

Alex and Rose are trying to factorise: 14x + 28

Alex says that the correct factorisation is: 7(2x + 4)

Rose says that the correct answer is: 2(2x + 14)

Who is right?

Rose

Alex

Neither of them

Fill in each blank with a number to factorise 8a + 12b - 4c completely.

Rose

Alex

Neither of them

Rory has factorised an expression and his answer is 3(3 − 7b).

What was the expression that he factorised?

3 - 7b

33 - 37b

21b - 9

9 - 21b

Factorise the following expression fully:

6x + 21

3(x + 21)

3(2x + 7)

3(x + 7)

3(2x + 21)

Which expression from the options below is equivalent to 3(2a + 8b)?

6a + 8b

6(a + 8b)

6a + 11b

6(a + 4b)

Factorise fully:

10x2 - 5

What should be written in the highlighted space above?

• Question 1

What is the highest common factor (HCF) of 24 and 18?

6
EDDIE SAYS
Let's start by listing our factors and comparing them. The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24, etc. The factors of 18 are: 1, 2, 3, 6, 9, 18, etc. The factor with the highest value which appears in both list is 6. Remember to look for the HCF in the upcoming questions.
• Question 2

Match each expression below to its correct, factorised form.

## Column B

5a + 15b
5(a + 3b)
25a - 5b
5(5a - b)
15a - 10b
5(3a - 2b)
15a - 20b
5(3a - 4b)
20a - 5b
5(4a - b)
EDDIE SAYS
This is a little tricky as every expression has a HCF of 5. The best way to approach this question is to carefully factorise each expression, one at a time. Let's have a look at a couple together. In 5a + 15b the common factor is 5, which goes outside of the bracket. What would go inside? 5 x a = 5a and 5 x 3b = 15b, so we'll have a and 3b inside of the bracket. So 5a + 15b = 5(a + 3) This question can be done in the same way: 25a - 5b. 5 x 5a = 25a 5 x -b = -5b So the factorised bracket will be: 5(5a - b) Can you check the final two independently?
• Question 3

Factorise:

4a - 12b

4(a - 3b)
4(a-3b)
4(a- 3b)
4(a -3b)
4 (a - 3b)
EDDIE SAYS
The HCF of 4 and 12 is 4, so 4 goes outside of the bracket: 4(__ _ __) We need to find a sum which gives a product of 4a first: 4 × a = 4a Then we need to find a sum that gives a product of -12b: 4 × -3b = -12 So our answer is: 4(a -3b)
• Question 4

Is it possible to factorise 8y + 3x?

No
EDDIE SAYS
8y + 3x cannot be factorised because 8 and 3 do not have any common factors, other than 1. Factorising by 1 doesn't help, as the outcome is no simpler than our original expression.
• Question 5

Alex and Rose are trying to factorise: 14x + 28

Alex says that the correct factorisation is: 7(2x + 4)

Rose says that the correct answer is: 2(2x + 14)

Who is right?

Neither of them
EDDIE SAYS
Both students are wrong here because neither of them used the HIGHEST common factor of 14 and 28 for their factorisation. Let's look at factors of 14 and 28. 14: 1, 2, 7, 14, etc. 28: 1, 2, 4, 5, 14, 28, etc. The HCF of 14 and 28 is 14. If we use this as our multiplier outside the brackets, we reach: 14(x + 2)
• Question 6

Fill in each blank with a number to factorise 8a + 12b - 4c completely.

EDDIE SAYS
There are three terms in this expression: 8a, 12 b and 4c. However, we can still factorise it by finding the highest common factor of 8, 12 and 4. The HCF of these three numbers is 4, so we put it outside of the bracket like this: 4(__ _ __ _ __) We then need to work out what goes inside of the bracket. We will have three terms in the brackets rather than two, but that's ok! The answer should be: 8a + 12b - 4c = 4(2a + 3b - c) Did you type the correct numbers into each blank?
• Question 7

Rory has factorised an expression and his answer is 3(3 − 7b).

What was the expression that he factorised?

9 - 21b
EDDIE SAYS
To check what expression Rory has factorised, we need to do the opposite operation, which is expanding the brackets. When we expand 3(3 - 7b) we reach 9 - 21b, so this is the expression that Rory factorised. Remember that factorising and expanding expressions are opposite operations.
• Question 8

Factorise the following expression fully:

6x + 21

3(2x + 7)
EDDIE SAYS
The highest common factor of 6 and 21 is 3. So 3 goes outside of our bracket: 3(__ _ __) But what goes inside of the bracket? We need to divide each term by 3 to find the answer: 3 × 2x = 6x 3 × 7= 21 2x and 7 go inside of the bracket.
• Question 9

Which expression from the options below is equivalent to 3(2a + 8b)?

6(a + 4b)
EDDIE SAYS
Did you try expanding the brackets first? 3(2a + 8b) = 6a + 24b None of our options match this so what should we do now? Let's try re-factorising this expression using a different factor. The highest common factor of 6 and 24 is 6, so let's put 6 outside of the brackets instead. 6 × a = 6a 6 × 4b = 24b Just put these together to reveal the correct answer: 6(a + 4b) Don't forget that different factorised expressions can give you the same expression when expanded.
• Question 10

Factorise fully:

10x2 - 5

What should be written in the highlighted space above?

- 1
-1
EDDIE SAYS
The number 5 has been placed on the outside of the brackets, so this is a factor of both the numbers inside the brackets. We need to reach -5 in the original expression, so what do we multiply 5 by to reach this? The answer is -1, as a negative and positive number multiply together to give a negative outcome. Fantastic work! Why not revise positive and negative numbers if you found you were rusty on this element of the activity at all?
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