The easiest case of polynomial division is when the divisor is a single term that happens to be a common factor of the dividend. You'll love when this happens. For the sake of the integer analogy, we'll do a couple of examples with integers first. You'll probably notice that we're not doing the integer problems in the most efficient way. That's not because we're being lazy or unsure of what we're doing. By approaching them in a certain manner, we'll be doing them in a way that will translate nicely to polynomials. Ah, there's a method to our madness...
Sample Problem
Write 
as a sum of two integers.
The fraction 
is the sum of the two fractions 
and 
. Since 
and 
, we have:
Sample Problem
Write 
as a sum of integers.
This is like the last problem, except we'll be splitting the fraction into a sum of four other fractions. Once you're done, try splitting an atom. Or your pants. Anything, really, as long as you're getting in some good splitting practice.
Performing division on each term, we find:
Now we'll do something similar with polynomials. Take five, integers. If the divisor is a common factor of the dividend, then the divisor divides (goes evenly into, with nothing left over) every term of the dividend, meaning we can break up the polynomial like we broke up the fractions above. Contrary to what you may have heard, breaking up isn't that hard to do.
Sample Problem
Find 
.
We can split up the terms of the dividend and rewrite this as 
.
Simplifying each term gives us the final answer: 4x + 8. Just cancel out those x's in the denominators.
Another way to look at this sort of problem is to factor. We'll let the integers be the opening act again. They did such a great job when they opened for Coldplay last weekend that they deserve to be given another opportunity.
Sample Problem
Write 
as a sum of two integers.
Pull out the common factor of 5 from the numerator and the denominator.
Then we can write the fraction as 
and cancel the 5 from the top and bottom to find 
.
Good job canceling the 5, but don't assume you'll never hear from it again. You're still on its email list.
After a bit of practice, you can skip the step of physically writing down where you pull the common factor out of each term...as long as you can do it mentally and still wind up with the correct answer, that is. If you can master this skill, who knows? You may be able to start doing all kinds of things mentally. Don't go all Chronicle on us, though.
When asked to divide a polynomial by a single term, you can use whichever method you like. You'll find the same answer whether you break up the polynomial first and then simplify the resulting terms, or pull out the common factor first. Don't do both at the same time, or you may disrupt the space-time continuum.