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Expanding brackets Expand and simplify Factors and multiples Powers and roots Algebraic expressions Adding and subtracting negative numbers Multiplying and dividing negative numbers Highest common factor Laws of indices Square numbers and square rootsThis topic is relevant for:

Here we will learn about factorising to single brackets by finding the highest common factor.

There are also factorising to single brackets worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

In order to factorise an algebraic expression into a single bracket:

- Find the highest common factor of each of the terms in the expression.
- Write the highest common factor (HCF) at the front of a single bracket
- Fill in each term in the bracket by multiplying out.

We can factorise lots of different types of expressions into single brackets including some quadratics like ^{2} + 5^{2} – 5x

For quadratic expressions of the form ^{2} + bx + c^{2} + bx + c

Check out our main factorising lesson for a complete summary of all the different ways we can factorise expressions, and then explore our other factorising lessons for detailed step-by-step guides and worksheets on each type.

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

DOWNLOAD FREEGet your free factorising single brackets worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEFully factorise:

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

1Find the highest common factor (HCF) of the numbers

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

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

\[\color{#FF9100}3(\qquad+\qquad)\]

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

What do I need to multiply

\[\color{#FF9100}3\times \color{#62F030}x = \color{#00bc89}3x\]

What do I need to multiply

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

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

We can check the answer by multiplying out the bracket!

\[3(x+2)=3x+6\]

Fully factorise:

\[\color{#7C4DFF}{14}\color{#00BC89}{-7}y\]

Find the highest common factor (HCF) of the numbers 14 and 7.

Factors of 14:

1, 14

2, 7

Factors of 7:

1, 7

The highest common factor (HCF) of

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

\[\color{#FF9100}7(\qquad-\qquad)\]

Fill in each term in the bracket by multiplying out.

What do I need to multiply

\[\color{#FF9100}7\times \color{#62F030}2 = \color{#7C4DFF}{14}\]

What do I need to multiply

\[\color{#FF9100}7\times \color{#92009E}y = \color{#00bc89}7y\]

\[\color{#FF9100}7(\color{#62F030}2-\color{#92009E}y)\]

*(don’t forget to keep the – here)*

We can check the answer by multiplying out the bracket!

\[7(2-y)=14-7y\]

Fully factorise:

\[\color{#FE47EC}8x^{2}\color{#00BC89}{+12}x\]

Find the highest common factor (HCF) of the numbers 8 and 12.

Factors of 8:

1, 8

2, 4

Factors of 12:

1, 12

2, 6

3, 4

Find the highest common factor (HCF) of the variables ^{2}

^{2} = x

The highest common factor (HCF) of ^{2}

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

\[\color{#FF9100}{4x}(\qquad+\qquad)\]

Fill in each term in the bracket by multiplying out.

What do I need to multiply ^{2}

\[\color{#FF9100}{4x} \times \color{#62F030}{2x} = \color{#FE47EC}8x^2\]

What do I need to multiply

\[\color{#FF9100}{4x} \times \color{#92009E}3 = \color{#00BC89}{12}x\]

\[\color{#FF9100}{4x}(\color{#62F030}{2x} + \color{#92009E}3)\]

We can check the answer by multiplying out the bracket!

^{2} + 12x

Fully factorise:

\[\color{#FE47EC}{15}y^{2}\color{#0EE2EF}{-10}xy\]

Find the highest common factor (HCF) of the numbers 15 and 10.

Factors of 15:

1, 15

3, 5

Factors of 10:

1, 10

2, 5

Find the highest common factor (HCF) of the letters ^{2}

^{2} = y ✕ y

The highest common factor (HCF) of ^{2}**and**

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

\[\color{#FF9100}{5y}(\qquad-\qquad)\]

Fill in each term in the bracket by multiplying out.

What do I need to multiply ^{2}

\[\color{#FF9100}{5y} \times \color{#62F030}{3y} = \color{#FE47EC}{15}y^2\]

What do I need to multiply

\[\color{#FF9100}{5y} \times \color{#92009E}{2x} = \color{#0EE2EF}{10}xy\]

\[\color{#FF9100}{5y}(\color{#62F030}{3y}\color{#92009E}{-2x})\]

We can check the answer by multiplying out the bracket!

\[5y(3y-2x)=15y^{2}-10xy\]

Fully factorise:

\[\color{#00BC89}{6}x \color{#00BC89}{+2}y \color{#7C4DFF}{- 12}\]

Find the highest common factor (HCF) of the numbers 6, 2 and 12

Factors of 6:

1, 6

2, 3

Factors of 2:

1, 2

Factors of 12:

1, 12

2, 6

3,4

The highest common factor (HCF) of

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

\[\color{#FF9100}2(\qquad+\qquad-\qquad)\]

Fill in each term in the bracket by multiplying out.

What do I need to multiply

\[\color{#FF9100}2 \times \color{#62F030}{3x} = \color{#00bc89}6x\]

What do I need to multiply

\[\color{#FF9100}2 \times \color{#92009E}y = \color{#00bc89}2y\]

What do I need to multiply

\[\color{#FF9100}2 \times \color{#F6C929}6 = \color{#7C4DFF}{12}\]

\[\color{#FF9100}2(\color{#62F030}{3x} \color{#92009E}{+y} \color{#F6C929}{-6})\]

*Don’t forget to keep the – here!*

We can check the answer by multiplying out the bracket!

\[2(3x+y-6)=6x+2y-12\]

Fully factorise:

\[\color{#0EE2EF}{12}xy\color{#FE47EC}{-4}x^{2}y\color{#FE47EC}{+8}xy^{2}\]

Find the highest common factor (HCF) of the numbers 12, 4 and 8

Factors of 12:

1, 12

2, 6

3,4

Factors of 4:

1, 4

2, 2

Factors of 8:

1, 8

2, 4

Find the highest common factor (HCF) of the letters ^{2}^{2}y^{2}

^{2}y = x ✕ x ✕ y

^{2} = x ✕ y ✕ y

The highest common factor (HCF) of ^{2}^{2}y^{2}

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

\[\color{#FF9100}4xy(\qquad-\qquad+\qquad)\]

Fill in each term in the bracket by multiplying out.

What do I need to multiply

\[\color{#FF9100}{4xy} \times \color{#62F030}3 = \color{#0EE2EF}{12}xy\]

What do I need to multiply ^{2}y

\[\color{#FF9100}{4xy} \times \color{#92009E}x = \color{#FE47EC}4x^2y\]

What do I need to multiply ^{2}

\[\color{#FF9100}{4xy} \times \color{#F6C929}{2y} = \color{#FE47EC}8xy^2\]

\[\color{#FF9100}{4xy}( \color{#62F030}3\color{#92009E}{-x}\color{#F6C929}{+2y})\]

*Don’t forget to keep the – here!*

We can check the answer by multiplying out the bracket!

\[4xy(3-x+2y)=12xy-4x^{2}y+8xy^{2}\]

These are some of the common misconceptions around factorising into single brackets

- We must
**fully**factorise12x ^{2}– 6x = 2(6x^{2}– 3x)

Here we have factorised the expression, however it is**not fully factorised**because 2 is not the highest common factor.6x is the highest common factor, so this is the correct final answer:12x ^{2}– 6x = 6x(2x – 1) 12x ^{2}– 6x

Even though this a quadratic expression we still factorise it to give a single bracket because it is not in the formax ^{2}+ bx + c

1. Fully factorise: 5x+10

5(x+10)

5x(x+2)

5(x+2)

x(5+10)

The highest common factor of 5 and 10 is 5 . We can divide each term in the original expression by 5 , which means the bracket must contain (x+2) .

2. Fully factorise: 8-2y

2(4-y)

2(4+y)

2y(4-y)

8(1-2y)

The highest common factor of 8 and 2 is 2 . We can divide each term in the original expression by 2 ; the bracket must contain a minus sign just like the original expression.

3. Fully factorise: 18x^{2}-12x

3x(6x-4)

6(3x^{2}-2x)

3(6x^{2}-4x)

6x(3x-2)

The highest common factor of 18x^{2} and 12x is 6x . We can divide each term by 6x , which means the bracket must contain (3x-2) .

4. Fully factorise: 20y^{2}+16xy

4y(5y-4x)

y(20y-16x)

2y(10y-8x)

4xy(5y-4x)

The highest common factor of 20y^{2} and 16xy is 4y . We can divide each term by 4y , which means the bracket must contain (5y-4x) .

5. Fully factorise: 18-6y+15x

3xy(6-2y+5x)

3(6-2y+5x)

18(1-2y+5x)

3(6+2y+5x)

The highest common factor of 18,6y and 15x is 3 . We can divide each term by 3 , which means the bracket must contain (6-2y+5x) .

6. Fully factorise: 12y-9x^{2}y+6y^{2}

12y(1-3x^{2}+2y)

3y(4+3x^{2}+2y)

3y(4-3x^{2}+2y)

3y(4-3x^{2}-2y)

The highest common factor of 12y, 9x^{2}y and 6y^{2} is 3y . We can divide each term by 3y , which means the bracket must contain (4-3x^{2}+2y) .

1. Factorise: 5x-20

Show answer

5(x-4)

(1 mark)

2. Factorise fully: 8x^{2} + 12xy

Show answer

4x(2x+3y)

(2 marks)

3. Factorise: x^{2} + 8x

Show answer

x(x+8)

(1 mark)

- Manipulate algebraic expressions by taking out common factors to factorise into a single bracket
~~Factorise quadratic expressions of the form x~~^{2}+ bx + c~~Factorise quadratic expressions of the form of the difference of two squares~~~~Factorising quadratic expressions of the form ax~~^{2}+ bx + c (H)

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