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 is no nicer way to write using fractions than . Similarly, there's no nicer way to write . 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 at least 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 set up the problem:
x – 7 ) x3 + 0x2 + 2x + 1.
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? 0 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. The final answer is
x3 + 2x + 1/x – 7 = x2 + 7x + 51 + 358/x – 7.
That is 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 . Since x2 + 7x + 51 is equal to
everything works out nicely. Huzzah!
If we work out using long division, we find 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