Factor 6x^2 + 12x

Date: 06/23/2003 at 13:33:04
From: Katie Huie
Subject: Math

Factor the following term:  4x-8

I don't understand how to factor this type of question.

Date: 06/28/2003 at 19:59:48
From: Doctor Ian
Subject: Re: Math

Hi Katie,

The idea of factoring is just to find things in common. We do this
sort of thing in English, actually. For example, instead of saying

  I went to England.  I went to France.  And I went to Sweden.

we would normally say

  I went to England, to France, and to Sweden.

or even

  I went to England, France, and Sweden.

What I did was find the parts that they all had in common, 

  I went 

or

  I went to

and I pulled them out front, so I just had to say it once.  Does that
make sense? 

Similarly, if I have something like 

  6x^2 + 12x

I can expand that to make all the multiplications visible:
 
  6*x*x + 12*x

Now, it turns out that 6 and 12 are themselves products of prime 
factors:

  2*3*x*x + 2*2*3*x

Looked at it this way, there's no reason to multiply by 2 twice, when
I can do it once, right?

  2*(3*x*x + 2*3*x)

Similarly, there's no reason to multiply by 3 twice, when I can do it
once:

  2*3*(x*x + 2*x)

And there's no reason to multiply by x twice when I can do it once:

  2*3*x*(x + 2)

So now I can simplify this to 

  6x(x + 2)

And that's factoring.  

Why would I _want_ to do something like this?  What's wrong with
the original form, 

  6x^2 + 12x ?

Nothing, really. But we don't often have expressions sitting around in 
isolation like this. Usually they're part of equations, and quite 
often the equations have zero on one side, e.g., 

  6x^2 + 12x = 0

Now, when you look at it in this form, it's not easy to see what
values of x will make the equation true. But when we write it in this
form, 

  6x(x + 2) = 0

we can see something interesting.  In order for the product of a bunch
of things to be zero, 

  this * that * other = 0

it has to be true that at least one of them is zero.  There's no other
way to get a product of zero, than to have zero as a factor. So when
we look at 

  6 * x * (x + 2) = 0

we can see right away that this will be true when x=0; and again when
x=-2.  (It can also be true when 6=0, but that rarely happens, so we
don't have to worry about it.) So this is one very powerful use of
factoring.  

Another is simplifying rational expressions. For example, if we have
something like 

   6x^2 + 12x
   ----------  
    -14 - 7x

we can factor the numerator and denominator to make it look like this:

    6x(x+2)     6x    (x+2)    6x         6x
   --------- = ---- * ----- = ---- * 1 = ----  
    -7(x+2)     -7    (x+2)    -7         -7

so long as x isn't equal to -2.  Which of these expressions would you
rather deal with?

Try factoring your original expression, and let me know what
you come up with. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/