So, recently I've been learning about partial fractions and their decomposition and use in Integration, however, I just don't understand why I should be decomposing some fractions in the way that they've told me to. Take the example
$$\frac{1}{(3x + 4)^5}.$$
To decompose this, I would have to convert this to
$$\frac{A}{3x + 4} + \frac{B}{(3x + 4)^2} + \frac{C}{(3x + 4)^3} + \frac{D}{(3x + 4)^4} + \frac{E}{(3x + 4)^5}.$$
Now, the next step would be to equate the numerators, but I don't see how this is possible when the denominators don't line up with each other, as the multiplication of the all the partial sum's denominators would be equal to $(3x + 4)^{15}$ and not $(3x + 4)^5.$