GCSE revision notes

It is important to understand the distinction between expanding and factorising.

These concepts occur across a range of Mathematical Ideas

Factorising – lots of examples

Read through each one carefully – make sure you understand each one

Factorising is about PUTTING an expression BACK INTO BRACKETS.

To understand factorising requires you to understand EXPANDING BRACKETS

Suppose you are asked to factorise: ,

 

We need to find the HIGHEST COMMON FACTOR (HCF) of 5 and 10.

 

The highest number which goes in to 5 and 10 is 5

 

So,

 

Factorise,

 

The HCF of 12 and 18 is 6

 

So

 

Factorise,

 

The HCF of 24 and 8 is 8  [ignore the minus sign for the moment

 

So,

 

Factorise,

 

This is more subtle, there is a common factor of

 

So,      [think carefully,

 

Factorise,

 

There are two parts to this

 

The HCF of 4 and 12 is 4

 

However, there is also a common factor of

 

So,

 

 

QUADRATIC factorising. Part 1:

If you are asked to factorise , we need to focus on the +6.

 

The KEY QUESTION here is: what two numbers MULTIPLY to make 6 and ADD to make 5

 

The numbers are

 

So,

 

If you are asked to factorise , we need to focus on the +16.

 

The KEY QUESTION here is: what two numbers MULTIPLY to make 16 and ADD to make 10

 

The numbers are

 

So,

 

Factorise,

 

So,

Factorise,

 

Be careful, we need two numbers which multiply to make

 

So,

Factorise,

 

So,

SPECIAL CASE: THE DIFFERENCE OF TWO SQUARES

 

Factorise,

 

NOTICE HOW 25 and  are both square numbers,  and

 

So,

Factorise,

So,

 

Factorise,

So,

 

Grade A factorising

It is much trickier to factorise expressions such as:

 

 

To attempt this requires a trial and error approach, focusing on the first and last term,

 

The terms which multiply to get  are  and an

 

To number which multiply to get 5 are 5 and 1

 

We can therefore list all the possibilities

 

 

 

WE THEN SEE WHICH ONE EXPANDS TO GIVE THE REQUIRED EXPRESSION

 

 

So,

AN EVEN LONGER EXAMPLE!!!

Factorise

 

The terms which multiply to make  are:        

 

The terms which multiply to make  are:           OR

 

WE COULD THEREFORE BEGIN TO LIST ALL THE POSSIBILITIES!!!

 

 

We can see how  expands to give

 

 

WOW, that takes some effort.

The next skill is to use factorising to solve quadratic equations.

There will be another set of notes on this area!