Factoring in Algebra

Factors

Numbers have factors:

And expressions (like x²+4x+3) also have factors:

Factoring

Factoring (called "Factorising" in the UK) is the process of finding the factors:

Factoring: Finding what to multiply together 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 is 2 × y
• 6 is 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 also the opposite of Expanding:

Common Factor

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

But to do the job properly 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.

So you could have:

3y²+12y = 3(y²+4y)

But we can do better!

3y² and 12y also share the variable y.

Together that makes 3y:

• 3y² is 3y × y
• 12y is 3y × 4

So you can factor the whole expression into:

3y²+12y = 3y(y+4)

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

More Complicated Factoring

Factoring Can Be Hard !

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!

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!

Experience Helps

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 might see 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

Yes!

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

Answer: 4x² - 9 = (2x+3)(2x-3)

How can you learn to do that? By getting lots of practice, and knowing "Identities"!

Remember these Identities

Here is a list of common "Identities" (including the "difference of squares" used above).

It is worth remembering these, as they can make factoring easier.

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)
a3+3a2b+3ab2+b3	=	(a+b)3
a3-3a2b+3ab2-b3	=	(a-b)3

There are many more like those, but those are the simplest ones.

Advice

The factored form is usually best.

When trying to factor, follow these steps:

"Factor out" any common terms
See if it fits any of the identities, plus any more you may know
Keep going till you can't factor any more

You can also use computers! There are Computer Algebra Systems (called "CAS") such as Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce and many more that are good at factoring.

More Examples

I said that experience helps, so here are more examples to help you on the way:

Example: w⁴ - 16

An exponent of 4? Maybe we could try an exponent of 2:

w⁴ - 16 = (w²)² - 4²

Yes, it is the difference of squares

w⁴ - 16 = (w² + 4)(w² - 4)

And "(w² - 4)" is another difference of squares

w⁴ - 16 = (w² + 4)(w + 2)(w - 2)

That is as far as I can go (unless I use imaginary numbers)

Example: 3u⁴ - 24uv³

Remove common factor "3u":

3u⁴ - 24uv³ = 3u(u³ - 8v³)

Then a difference of cubes:

3u⁴ - 24uv³ = 3u(u³ - (2v)³)

= 3u(u-2v)(u²+2uv+4v²)

That is as far as I can go.

Example: z³ - z² - 9z + 9

Try factoring the first two and second two separately:

z²(z-1) - 9(z-1)

Wow, (z-1) is on both, so let us use that:

(z²-9)(z-1)

And z²-9 is a difference of squares

(z-3)(z+3)(z-1)

That is as far as I can go.