Repeated Linear Factors in denominator of fraction e.g. $\frac{2x^2 + 2x + 18}{x(x-3)^2}$ I have the following fraction below and I must find the partial fraction decomposition.
$$\frac{2x^2 + 2x + 18}{x(x-3)^2}$$
Now, I thought I could simplify this into the following... 
$$\frac{2x^2 + 2x + 18}{x(x-3)^2} = \frac{A}{x} + \frac{B}{(x-3)^2}$$
However, my professor said that this is incorrect, and instead it should be written as:
$$\frac{2x^2 + 2x + 18}{x(x-3)^2} = \frac{A}{x} + \frac{B}{x-3} + \frac{C}{(x-3)^2}$$
I'm confused. 
I can't seem to find an explanation for why my method is wrong and why my professor's solution is correct, using algebraic reasoning. 
Is there a reason why this is the case?

Consider - 
$$\frac{2x^2 + 2x + 18}{x(x-3)^2} = \frac{A}{x} + \frac{Bx + C}{(x-3)^2} = \frac{A}{x} + \frac{Bx}{(x-3)^2} + \frac{C}{(x-3)^2} =
\frac{A}{x} + \frac{B'}{x-3} + \frac{C}{(x-3)^2}$$
Where B is a constant that can be modified. 
Hence the professor's solution is correct.
