Single brackets: Expanding and factorising

In a nutshell

There are two main operations to do with single brackets, expanding them and factorising into a single bracket. When expanding single brackets multiply each term within the bracket separately. When factorising into a single bracket take out the biggest factor and highest power first.

Expanding single brackets

When expanding a bracket the number or letter outside of the bracket must be multiplied by all terms within the bracket. 

PROCEDURE

1. Multiply the number and letter outside of the bracket by the number and letter in the first term inside the bracket.
   
2. Multiply the number and letter outside of the bracket by the number and letter in the second term inside the bracket.
   
3. Then add both products together to get your final answer.
   

Note: When there is no number in front of a letter, it is 1.

Example 1

Expand the following bracket

​$$2x(2x - 4)$$

Multiplying the first term inside the bracket with the term outside the bracket gives us

$$2x \times 2x = 4x^2$$​​

Multiplying the second term inside the bracket with the term outside the bracket gives us

$$2x \times -4 = -8x$$​​

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Then adding both products together gives us

$$4x^2 + (-8x)$$​​

Therefore the final answer is

$$\underline {2x(2x-4) = 4x^2- 8x}$$​​

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Example 2

Expand the following bracket

​$$-a(3 + 5b)$$

​By multiplying the first term inside the bracket with the term outside the bracket this gives us

​$$-a \times 3 = -3a$$​​

​By multiplying the second term inside the bracket with the term outside the bracket this gives us

$$-a \times 5b = -5ab$$​​

Then adding both products together gives us

$$\underline {-3a + (-5ab) = -3a -5ab}$$​​

Note: When there is a minus sign in front of the term outside the bracket, it changes the sign inside the bracket.

Factorising into a single bracket

This is the exact opposite of expanding a single bracket. Taking out factors for numbers and letters makes this easier.

PROCEDURE

1. ​Find the highest common factor (HCF) of both terms.
   
2. Take out powers of the letters in both terms.
   
3. Put a bracket around the remainder of the two terms, and write the HCF and powers taken out in one term outside of the bracket.
   

Note: To check if the answer is correct simply expand your answer and it should be the same as the original question.

Example 1

Factorise the expression

$$2x^2 + 8x$$

The HCF of both terms is $$2$$. The powers of letters that can be taken out of both terms is $$x.$$ The HCF and powers of letters to be taken out combined is

$$2x.$$

Therefore the final answer is

​$$\underline {2x^2 + 8x = 2x(x+4)}$$

Example 2

Factorise the expression

​$$15ab + 50b$$

The HCF of both terms is $$5.$$ The powers of letters that can be taken out of both terms is $$b.$$ The HCF and powers of letters to be taken out combined is $$5b.$$​

Therefore the final answer is

$$\underline {15ab +50b =5b(3a + 10)}$$​​