TITLE: Easy ways to find partial fraction representation? (via a concrete example)
QUESTION [8 upvotes]: In a homework assignment (about generating functions) the students find themselves having to expand $\frac{3-7x+9x^{2}-3x^{3}}{\left(1-x\right)^{4}}$ intro partial fractions. Using some automated tool (e.g. Wolfram alpha) the expansion is found right away: $\frac{3}{1-x}-\frac{2}{\left(1-x\right)^{3}}+\frac{2}{\left(1-x\right)^{4}}$.
However, sending the students to use Wolfram alpha instead of computing by themselves seems problematic to me. However, I have no idea how the partial fraction representation can be found without a lot of dirty work involved, which will ensure the students hate the exercise instead of learning from it.
My question - is there some "nice" way of doing such things I am simply unaware of? Of course, the best answer is a demonstration on the concrete case of $\frac{3-7x+9x^{2}-3x^{3}}{\left(1-x\right)^{4}}$ - but only give it if your method is simple enough so that writing it down is not a lot of work...

REPLY [6 votes]: In this particular case, I think the quickest way is to introduce a new variable $t=1-x$. So just substitute $x=1-t$ and expand:
$$
\frac{3-7x+9x^{2}-3x^{3}}{\left(1-x\right)^{4}}=
\frac{3-7(1-t)+9(1-t)^{2}-3(1-t)^{3}}{t^{4}}=
\frac{2-2t+0t^2+3t^3}{t^4}=
\frac{2}{t^4} - \frac{2}{t^3} + \frac{3}{t}.
$$
Now put $t=1-x$ back, and you have the partial fractions decomposition.