ah okay, sorry about that, I misunderstood. so all the greatest coomon factors are 12, (4-3x^3)^3, (1-2x)^5?
There we go. Now, in all this business of finding the greatest common factor, let's not forget that we're trying to simplify the original expression. Now that you have the greatest common factor, what you do is factor that out of both terms. What do you get?
No. You need to factor out the greatest common factor from each term. Once you do that, what's left? For example, if I have 36 and 100, we figured out that the greatest common factor was 4. Hence, let's say we wanted to find the sum 36 + 100. We could write it as
36 + 100 = 4 * 9 + 4 * 25 = 4 ( 9 + 25 ).
That's what you're trying to do here, only with polynomials.
So what do you get?
Ok, here's what you need to do. You've got your greatest common factor, we'll call it x. You've got two terms, we'll call them xy and xz. You have the expression xy + xz, and you want to factor out the greatest common factor x thus: x(y+z). The problem is, you know what xy is, and what xz is, but you don't know what y and z are. How do you find out? By division! That is, y = (xy)/x, and z = (xz)/x. So take each term and divide out the greatest common factor, and that'll be what you have left inside the parentheses. Make sense? Let me give you a slightly more complicated example: simplify
$\displaystyle (x-1)^{4}(x+3)^{2}+(x-1)^{3}(x+3)^{5}.$
The greatest common factor is
$\displaystyle (x-1)^{3}(x+3)^{2}.$
Dividing through by this yields the following:
$\displaystyle \frac{(x-1)^{3}(x+3)^{2}}{(x-1)^{3}(x+3)^{2}}\left[(x-1)^{4}(x+3)^{2}+(x-1)^{3}(x+3)^{5}\right]$
$\displaystyle =(x-1)^{3}(x+3)^{2}\left[\frac{(x-1)^{4}(x+3)^{2}}{(x-1)^{3}(x+3)^{2}}+\frac{(x-1)^{3}(x+3)^{5}}{(x-1)^{3}(x+3)^{2}}\right]$
$\displaystyle =(x-1)^{3}(x+3)^{2}\left[(x-1)+(x+3)^{3}\right].$
You could multiply out what's in the square brackets if you choose. So what I did here was to multiply by a fancy way of writing 1 (the thing outside the bracket), then distribute the denominator only into each term inside the parentheses, and then perform the divisions. Make sense?
thank you for being patient with me, I am getting so close. now for whatever reason the problem is with the (-9x^2)
so the (-9x^2) and the 4 make the -36 and the -12 comes for 6(-2) correct? and that is where I get the greatest common factor of 12 for those two, correct? then I divide out the way you showed me, but then there's an (x)^2 left over.