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Here's one freaky thing that can happen -- and it happens a lot:

Look at this guy:

( x^3 - 8 ) / ( x - 2 )

Whenever you set these guys up, you need two things:

1) all the powers in descending order: x^5, x^4, x^3, x^2, x, #

2) All the powers represented: For x^3 - 8, we have no x^2 guy and no x guy... So, we need to write it as x^3 + 0x^2 + 0x - 8
We haven't changed his value... But, we've added some important place-holders. I call them "dummy guys."

If you don't have the dummy guys in the setup, you're going to get stuck with a situation that you don't know how to deal with...