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In order to access this I need to be confident with:

Expanding brackets Expand and simplifyFactors, multiples, powers and roots

Algebraic expressionsAdding and subtracting negative numbers

Multiplying and dividing negative numbers

Highest common factor (HCF)

Laws of indicesSquare numbers and square roots

This 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~~ + x

~~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} = x ✕ ~~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}

~~x~~ ✕ ~~y~~

^{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

Show answer

= 5(x + 2)

2. Fully factorise: 8 – 2y

Show answer

= 2(4 – y)

3. Fully factorise: 18x^{2} – 12x

Show answer

= 6x(3x – 2)

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

Show answer

= 4y(5y – 4x)

5. Fully factorise: 18 – 6y + 15x

Show answer

= 3(6 – 2y + 5x)

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

Show answer

3y(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)

- Algebraic expressions
- Solving equations
- Quadratic equations

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