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So far, when we've been doing these long division things, we've gotten a nice 0 at the end.

0

This means that the divisor polynomial went into the other guy cleanly -- no remainder.

As you can imagine, things only work out nicely like this when math teachers design them this way!

So, we're going to need to learn to deal with remainders.

Let's do a really basic review:

15 / 3 = 5 ... no remainder, 15 / 7 with a remainder of 1

For the second one, we're really doing this:

15 / 7 gives 2 ... multiplying gives 14 and subtracting gives 1 ... 1 is the remainder

We officially write it like this:

15 / 7 = 2 1/7 ... 1 is the remainder and 7 is the original divisor

Remember? Well, it works the same way with polynomials.