FACTORIZATION

Factoring in Algebra

Factors

Numbers have factors:

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

Factoring

Factoring 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:

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!

 

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

Example: Factor 4x² − 9

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

But knowing the Special Binomial Product gives us a clue called 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)

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

Examples

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)

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)

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)