# Factoring out two terms

• Jul 3rd 2012, 12:11 AM
astuart
Factoring out two terms
Hello,

I'm having trouble understanding how two terms are to be factored out in an equation. By my textbook, I have...

(1+x2)2(-2x)-(1-x2)2(1+x2)(2x) / (1+x2)4

According to the textbook, I have to factor out like this

(1+x2)(2x) [(1+x2)(-1)-2(1-x2)] / (1+x2)4

Anyway, silly question, but how exactly am I supposed to factor out with both 1+x^2 and 2x? Any tips?
• Jul 3rd 2012, 01:07 AM
rowe
Re: Factoring out two terms
Assuming:

$\displaystyle \frac{{(1+x^2)}^2(-2x) - (1-x^2)2(1+x^2)(2x)}{{(1+x^2)}^4}$

Notice that in the numerator, there are two terms:

$\displaystyle {(1+x^2)}^2(-2x)$

and $\displaystyle (1-x^2)2(1+x^2)(2x)$

that subtract from each other. We need to find common factors in both sides of the subtraction that we can "pull out". Let's look at the factors in the numerator like this:

$\displaystyle a = (1+x^2)$
$\displaystyle b = (2x)$
$\displaystyle c = (1-x^2)$

Then we have this:

$\displaystyle (a^2 \cdot -b) - (c \cdot 2ab)$

You should be familiar enough with extracting common factors to know that this is:

$\displaystyle ab((a-1)-2c)$

Now we know that our factors are a & b, which means in the original you can extract $\displaystyle (1+x^2)$ and $\displaystyle 2x$ from both terms. You might be able to see how one of these factors will cancel out with a factor in the denominator, too. If this is still confusing, I suggest you try more simple factoring exercises.