Question

find the partial fraction decomposition of the rational expression (3x^2-x-14)/(x^3-x)

Answer 1

Numerator is of lower degree than the denominator, ok good that takes care of the first step.

Next, we need our denominator factored as far as possible.
Do you understand how to factor (x^3-x) ?

Answer 2

\[\begin{align*}\frac{3x^2-x-14}{x^3-x}&=\frac{3x^2-x-14}{x(x-1)(x+1)}\\\\
&=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x+1}\\\\
3x^2-x-14&=A(x-1)(x+1)+Bx(x+1)+Cx(x-1)\\
&=(A+B+C)x^2+(B-C)x-A
\end{align*}\]
Match up the coefficients, you have
\[\begin{cases}A+B+C=3\\
B-C=-1\\
-A=-14\end{cases}\]
Solve the system...