When using the method of partial fractions how do you choose what type of numerator to use and how do you know how many partial fractions there are?

There are as many fractions as there are factors of the denomenator.  If the denomenator of one of the partial fractions is linear then the numerator is a constant. If the denomenator is a linear term squared then the numerator is once again a constant. However, if the denomenator is a quadratic term then the numerator will be linear.