If you take a quick gander at the table on this page: http://tutorial.math.lamar.edu/Classes/CalcII/PartialFractions.aspx
You'll notice there are some rules about what the form of a partial fraction expansion should be, based on the factors of the original rational function.
At this point I have learned partial fractions at school more than 3 times, but no one bothers to explain the reasoning behind these rules. Now I'm frustrated because every time I do it myself, I can never remember how to do it right and end up looking up the table on the linked page above.
So my question is, how do you "derive" these rules for the form of a partial fraction expansion?