Factoring in Algebra


Numbers have factors:

factors 2x3=6

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



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 we 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:

expand vs factor

Common Factor

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

But to do the job properly we need the highest common factor, including any variables

Example: factor 3y2+12y

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

So we 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 we 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 we have to figure what got multiplied to produce the expression we are given!


factoring cake
It is like trying to find which ingredients
went into a cake to make it so delicious.
It can be hard to figure out!

Experience Helps

With more experience factoring becomes easier.

Example: Factor 4x2 − 9

Hmmm... there don't seem to be any common factors.

But knowing the Special Binomial Products gives us a clue called the "difference of squares":

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



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.

factor expand
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 most useful ones.


The factored form is usually best.

When trying to factor, follow these steps:

There are also Computer Algebra Systems (called "CAS") such as Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce and others that can do factoring.

More Examples

Experience does help, 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:


And z2−9 is a difference of squares


That is as far as I can go.

Now get some more experience:

338, 339, 2047, 2048, 2049, 178, 2050, 3177, 3178, 3179