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Now that we know what rational expressions are, we can investigate general polynomial division. Any polynomial division problem can be written as a rational expression, and any rational expression can be interpreted as a polynomial division problem. For example, could be read as the rational expression "8x + 9 over 3x2," or it could be read as the polynomial division "8x + 9 divided by 3x2." That fraction line is pulling double-duty. We hope it's compensated accordingly.
To do polynomial division when we don't have the guarantee that things will work out evenly, we use the same techniques we used earlier for dividing polynomials. We also cross our fingers and hope that it'll work out evenly anyway.
There's also a neat trick we can sometimes use if we want to feel clever. Yes, even more clever than usual.
The first thing to do when dividing polynomials is to look for and remove any common monomial factors. In terms of rational expressions, "looking for common factors" means that we're looking for those that go into every term in the numerator and into every term in the denominator. These are the factors we can pull out and cancel. At least it's way easier than canceling your LA Fitness membership. Those people are relentless.
The difference between what we did with common factors earlier and what we're doing now is that now we may cancel a common factor and still have a rational expression instead of a polynomial. That's fine. We can live with that, right? We've had more than our fill of polynomials lately anyway.
What's the simplified version of the rational expression ?
After some hunting, we can see that this guy has a common factor of x2 in the numerator and denominator. There are x's all over the place, but this is the most we can grab out of every single term. We pull this factor out of both the numerator and denominator.
And then we can cancel x2 from the top and bottom.
That's as simple as it gets this time. We've still got a rational expression on our hands, but at least it's a bit less intense-looking.
After looking for single-term common factors, the next thing to do is to factor the numerator and denominator to see if there are any other factors we can cancel. We're on an absolute canceling tear. We can't be stopped!
Okay, fine. If it looks like factoring the numerator and factoring the denominator are reasonable tasks, go for it. Otherwise, skip this step. We're on an absolute unreasonable step-skipping tear. We can't be stopped!
Simplify the rational expression as much as possible.
First off, our expression factors as . We can cancel out a factor of (x + 1) from the numerator and denominator, which leaves us with .
Meanwhile, the expression is way too grody to factor. Our eyes are burning from looking at it. However, it turns out to factor quite nicely:
But there's no way to know this from looking at it unless you're some kind of mathematical savant. If you are, what in the world are you doing here? Don't you have bigger fish to fry?
Once we've canceled all the factors we can, we write out the new numerator and denominator (the unfactored versions) and look at their degrees. If it's too many degrees, we'll stay inside and eat cheese puffs.
If the degree of the numerator is smaller than the degree of the denominator, there's nothing more to do. Think about fractions. There's no nicer way to write using fractions than . Similarly, there's no nicer way to write , since the top has a degree of 1 but the bottom is rocking a degree of 4. Of course, we could always put a "please" on the end of it, but this is one instance where manners don't matter. We're sure your parents would agree.
If the degree of the numerator is greater than or equal to the degree of the denominator, then we can use long division. This will work almost exactly like it did earlier, except that now we can have non-zero remainders. Well, it was nice while it lasted.
First, remember how this works with integers.
If we want to divide 100 by 3 using long division, it looks like this:
We stop at 1, because 3 doesn't fit into 1. To write the final answer, we stick the remainder over the divisor and write:
Our remainder is converted into a fraction. What a flip-flopper.
In this example, we found that the quotient of two integers was a rational number. Sometimes when we find the quotient of two polynomials, we'll get a rational expression as our final answer. With polynomials, we continue the long division until the degree of the remainder is less than the degree of the divisor. Or until we start experiencing severe hand cramps; same difference.
We work this out like the long division problems we did earlier.
First we see how many times x goes into x3:
Then we see how many times x goes into 7x2:
Then how many times x goes into 51x:
Now we're a bit stuck, because x doesn't "fit" into 358. Trust us. We tried, but then it got stuck and we had to grease that sucker to get it back out again.
In other words, the degree of 358 (which is 0) is less than the degree of x – 7 (which is 1). Right? Zero was less than 1 the last time we checked. Yep, still is. We're done with the long division part, and we have a remainder of 358. So we stick that remainder in a fraction over the divisor to get our final answer:
That's one top-heavy remainder, but it's all we can do, thanks to our degree differential. Yes, we're blaming it on the degrees. Mostly because we don't have a dog to blame it on.
We can check this answer by showing that is equal to . First we turn x2 + 7x + 51 into a fraction:
Then we add and simplify:
Everything works out nicely. Huzzah!
If we work out using long division, we get a quotient of 2x5 + 7x4 + 11x3 + 16x2 + 24x + 36 with a remainder of 108. Sheesh. We sure hope that whole thing fits on our stationery.
The final answer is:
First, you take the rubber ball in your left hand, then tap the back of that hand with the wand held in your right. Say a few magic words, and then...oh. This isn't amateur magician hour? Our bad.
We still have a good trick for you to try, though. It may not impress your baby brother or Grandma Jeannie, but it will impress the person grading your algebra problems, and that's a swell magic trick in its own right.
When we divided polynomials by monomials—not polynomials that have contracted a viral infection, in case you were wondering—we split up the terms of the polynomial and divided each term by the monomial. We can still split up the terms of the polynomial even if we're dividing by another polynomial, except we'll need to do it in a clever way. We also shouldn't wear mismatched socks while we do this, since it damages our cleverness rating.
Split the fraction up into two fractions. Peel it apart like you're getting ready to eat some string cheese.
Now the first fraction is only a cleverly disguised version of 1. The second fraction can't be made any nicer, since the degree of the numerator is less than the degree of the denominator. At least we're not stuck with something like Kevin Bacon6.
That one can get complicated.
Because we can't do anything more with the second fraction, here's our final answer:
By carefully splitting up polynomials and using the fact that adding 0 to an expression doesn't change the value of the expression (it also doesn't change the amount of our allowance, Dad), we arrive at another way to do polynomial division. By the way, we can also add anything equivalent to 0, such as x – x or -4y + 4y, to an expression without changing the value of the expression. This will be useful because you can bend, twist, and manipulate these terms until they're exactly what you need. They're like Silly Putty in your hands, but without that nasty after-smell.
This is almost like our first example. If only we had 4x instead of 3x on the top, we would know what to do. Maybe if we close our eyes and hope really hard...nope. Still there. We're starting to wonder if that ever works in real life.
We could turn 3x into 4x by adding one copy of x, but that would change the value of the expression, not to mention the fact that FedEx Office's prices have gotten truly outrageous. It would probably cost us $20 for that one copy. However, we can add a clever form of 0 to get 4x in the numerator without changing the value of the expression. If we were any more devious, we'd probably be arrested.
By putting the 3x and the x together, we can rewrite this expression as:
Ta-da! Now we have something we can split up.
Since the degree of -x – 2 is less than the degree of x2 + 4x, we're done. Let's pop open a bottle of Sunny D and celebrate.