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Factoring - Introduction

In Algebra "Factoring" (called "Factorising" in the UK) means finding what you can multiply to get an expression.

It is like "splitting" an expression into a multiplication of simpler expressions

Example: factor 2y+6

Both 2y and 6 have a common factor of 2:

2y = 2 × y

6 = 2 × 3

So you can factor the whole expression into:

2y+6 = 2(y+3)

So, 2y+6 has been "factored into" 2 and y+3

Factoring is the opposite of Expanding:

expand vs factor

Common Factor

In the previous example we saw that 2y and 6 had a common factor of 2

But, to make sure that you have done the job properly you need to make sure you have the highest common factor, including any variables

Example: factor 3y2+12y

Firstly, 3 and 12 have a common factor of 3.

But 3y2 and 12y also share the variable y.

Together that makes 3y:

3y2 = 3y × y

12y = 3y × 4

So you can factor the whole expression into:

3y2+12y = 3y(y+4)

 

Check: 3y(y+4) = 3yy + 3y × 4 = 3y2+12y

More Complicated Factoring

The examples have been simple so far, but factoring can be very tricky.

Because you have to figure what got multiplied to produce the expression you are given!

factoring cake It can be like trying to find out what ingredients went into a cake to make it so delicious. It is sometimes not obvious at all!

But the more experience you get, the easier it becomes.

Example: Factor 4x2 - 9

Hmmm... I can't see any common factors.

But if you know your Special Binomial Products you could recognise it as the "difference of squares".

Because 4x2 is (2x)2, and 9 is (3)2,

so we have:

4x2 - 9 = (2x)2 - (3)2

And that can be produced by the difference of squares formula:

(a+b)(a-b) = a2 - b2

Where "a" is 2x, and "b" is 3.

So let us try doing that:

(2x+3)(2x-3) = (2x)2 - (3)2 = 4x2 - 9

It works!

 

So the factors of 4x2 - 9 are (2x+3) and (2x-3):

Answer: 4x2 - 9 = (2x+3)(2x-3)

Here are some expressions, and the factors that produced them:

a2 - b2  =  (a+b)(a-b)
a2 + 2ab + b2  =  (a+b)(a+b)
a2 - 2ab + b2  =  (a-b)(a-b)
a3 + b3  =  (a+b)(a2-ab+b2)
a3 - b3  =  (a-b)(a2+ab+b2)

There are many more like those!