<s>[INST] How to Decompose into Partial Fractions Why is it useful to write:
$$\frac{}{(x+1)^2(x-1)}=\frac{}{(x+1)^2}+\frac{}{x+1}+\frac{}{x-1}$$
and not:
$$\frac{}{(x+1)^2(x-1)}=\frac{}{(x+1)^2}+\frac{}{x-1}$$
when decomposing into partial fractions?
 [/INST] For example, try decomposing $\frac{1}{(x+1)^2(x-1)}$ using the second form:
\begin{align}
    \frac{1}{(x+1)^2(x-1)} &= \frac{A}{(x+1)^2} + \frac{B}{x-1} \\
    \frac{1}{(x+1)^2(x-1)} &= \frac{A(x-1) + B(x+1)^2}{(x+1)^2(x-1)}.
\end{align}
Then $B$ must be zero by looking at the $x^2$ term, and then $A$ must be zero as well. So as the commenter pointed out, it just doesn't work. But using the first form we get
$$\frac{1}{(x+1)^2(x-1)} = -\frac{1}{2 (x+1)^2}-\frac{1}{4 (x+1)}+\frac{1}{4 (x-1)}.$$
</s>