Partial Fractions: Solving Numerator Issues

[Mentallic]

I'm having trouble understanding what the numerator needs to be in the partial fractions.
e.g.

$$\frac{1}{(x-1)(x-2)^2}\equiv \frac{A}{x-1}+\frac{Bx+C}{(x-2)^2}$$

Notice how the first numerator has a constant A, while the second is linear Bx+C.
Actually... just now I think I may understand it. Does it have to do with the fact that during synthetic division, the remainder is always 1 degree less than the divisor? The second fraction's denominator is a quadratic, so its numerator should be linear?

 

[statdad]

"Does it have to do with the fact that during synthetic division, the remainder is always 1 degree less than the divisor? The second fraction's denominator is a quadratic, so its numerator should be linear?"

Essentially - the numerator should always be the "most general" polynomial of lower degree than the denominator