<s>[INST] How to solve this partial fraction decomposition? Please help me to solve the following partial fraction decomposition:
$$\frac{1-v^2}{v+v^3} = \frac{A}{v}+\frac{Bv+C}{1+v^2}$$
 [/INST] The usual method would be to multiply both sides of this equation by $v+v^3$:
$$\begin{align}\frac{1-v^2}{v+v^3} &= \frac{A}{v}+\frac{Bv+C}{1+v^2}\\
1-v^2 &= A(1+v^2) + (Bv+C)(v)\\
1-v^2 &= Av^2 + A + Bv^2 + Cv\\
-v^2 + 0v + 1 &= (A+B)v^2 + Cv + A
\end{align}$$
At this point, you can equate coefficients on the two sides. Since the $v^2$ coefficients must match, we have $-1=A+B$. Since the $v$ coefficients must match, we have $0=C$, and since the constant terms must match, we have $1=A$.
Do you see how that's working?
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