Step 1 :The given rational expression is a proper fraction, so we can directly proceed to write the partial fraction decomposition.
Step 2 :The denominator factors as \(x(x-3)(x+3)\), so the partial fraction decomposition will have the form \(\frac{A}{x} + \frac{B}{x-3} + \frac{C}{x+3}\).
Step 3 :We can find the values of A, B, and C by equating the original rational expression with the partial fraction decomposition and solving the resulting system of equations.
Step 4 :The solution to the system of equations gives us \(A = -\frac{2}{9}\), \(B = \frac{5}{18}\), and \(C = -\frac{1}{18}\).
Step 5 :Substituting these values back into the partial fraction decomposition gives us the final answer.
Step 6 :\(\boxed{\frac{-2}{9x} + \frac{5}{18(x-3)} - \frac{1}{18(x+3)}}\)