Originally Posted by

**mathDad** Typically, if the numerator's degree is more than the denominator, I think you do polynomial division, first (right?). But what if that is not convenient (or if we're just too lazy to distribute). Say, for example you want to partial fraction decompose:

$\displaystyle \frac{x^4}{(x-1)^3}$

I could FOIL the bottom, but we want to find if there is another way to do this. We tried

$\displaystyle \frac{x^4}{(x-1)^3} = \frac Ax + \frac B{x^2} + \frac C{x^3} = \frac {Ax^2 +(-2A+B)x +(A-B+C)}{(x-1)^3}$.

But, there are no $\displaystyle x^4$ terms on the right and so it ends up with A, B, and C all equal to 0 (not surprisingly).

So is the only way to do this to expand the bottom? What if we had to do partial fractions on $\displaystyle \frac{x^9}{(x-1)^8}$ (or bigger)? How do you partially fraction decompose that (sans a CAS)?