TITLE: Question about partial fractions with irreducible quadratic factors
QUESTION [12 upvotes]: Given this rational function: $$\frac{-4x^4-2x^3-26x^2-8x-44}{(x+1)(x^2 +3)^2}$$
The decomposition would look like this: $$\frac{A}{x+1} + \frac{Bx+C}{(x^2+3)} + \frac{Dx+E}{(x^2+3)^2}$$
And the final answer would be: $$\frac{-4}{x+1} - \frac{2}{(x^2+3)} - \frac{2}{(x^2+3)^2}$$
But, if you were to set it up like this: $$\frac{A}{x+1} + \frac{B}{(x^2+3)} + \frac{C}{(x^2+3)^2}$$
You end up with the same answer:
$$\frac{-4}{x+1} - \frac{2}{(x^2+3)} - \frac{2}{(x^2+3)^2}$$
I want to know why this is. 
I thought that for irreducible quadratic factors, you needed a linear term in the numerator. Is this purely coincidence that it worked with constant terms in the numerator?

REPLY [4 votes]: I agree with the other answers that in some sense it is a coincidence, but I wouldn't say that it is just a coincidence. If this came from a textbook, then the author of the textbook started with the answer, which happens to have constant terms. If the book in question is a calculus textbook and the intention of the problem is to integrate the original function by doing a partial fraction decomposition, then the problem was probably rigged in advance so that neither the algebra nor the subsequent integration are too onerous for a homework problem. These considerations bias the problems, so that solutions with constant numerators appear in textbooks more often than they would otherwise be expected to. If thought of as a problem in reverse-engineering a textbook (or exam) problem, a reasonable heuristic might be to first see if there is a solution with constant terms, and only if and when that heuristic fails, search for a more general solution. A danger of following such a heuristic is that a grader might nevertheless penalize you since most students who use that heuristic don't understand that it is just a heuristic.