I have trouble with the logic behind Partial Fractions, which I will elaborate about here. I have two problems with it.
Firstly, why, if I have a polynomial like
$$f(x)= \frac{3}{(x+4)^2(x+3)}$$
That when I decompose it I get something like:
$$\frac{3}{(x+4)^2(x+3)} = \frac{A}{x+4} + \frac{B}{(x+4)^2} + \frac{C}{x+3}$$
But why do I have an $x+4$ and a $(x+4)^2$ when there's only a $(x+4)^2$ and $x+3$ in the denominator? It seems like there is a lack of equivalence here, like we are breaking a conservation law here.
For instance, if it were like
$$\frac{3}{(x+3)(3x+4)}$$
We would have, more intuitively in my opinion:
$$\frac{3}{(x+3)(3x+4)} = \frac{A}{x+3} + \frac{B}{3x+4}$$
Which makes me feel like the logical proceed of the above example, and applying it to the first would be this:
$$\frac{3}{(x+4)^2(x+3)} = \frac{A}{(x+4)^2} + \frac{B}{x+3}$$
Okay, secondly:
Why, when we're dealing with a term of the form $(x^2 + C)$ where $C$ is some constant, its partial fraction decomposition is of the form $Ax + B$ in the numerator, like in the following example:
$$\frac{4}{(x-4)^2(x^2+3)} = \frac{A}{x-4} + \frac{B}{(x-4)^2} + \frac{Cx + D}{x^2+3}$$
I'm not sure what clues one in to doing that?