Partial Fractions: What if the top has a higher power than the bottom?

Partial fractions decomposition only works when the numerator has a smaller degree than the denominator.  For example, here:

the numerator has degree 2 (because of the s-squared), and the denominator has degree 1, so partial fractions won’t work. What do we do? We need to divide the top by the bottom, using polynomial long division (this is another trick you may or may not remember from Algebra).  When we are done, we get:

and we can proceed to take the Inverse Laplace Transform of the expression on the right.

To see how long division works for polynomials, check out these videos:

Basic examples:

Another example:

Best of luck – write back if you get stuck.
-Prof. Reitz