# Partial fraction decomposition not working

While trying to do partial fraction decomposition on $$\frac{x^4 +1}{x(x^2+1)^2}$$ I first equated it to $$\frac{A}{x}+\frac{Bx+1}{x^2+1}+\frac{C}{(x^2+1)^2}$$

On solving this, by adding the fractions, you get $$\frac{x^2(A+B)+A+x}{x^3+x}+\frac{C}{x^4+2x^2+1}$$ so obviously $$A$$ has to be equal to $$1$$ for the numerator's constant to be 1, and similarly, going forward, $$B=-1$$ and $$C$$ is -2, but when you write it out, you get an extra $$x^5+2x^2+x$$ in the numerator.

As far as I know, the method works everywhere. Can someone tell me where I've messed up here?

• You assumed the numerator in the middle summand to be a linear polynomial, so why the last one term only has a constant numerator?
– xbh
Apr 4 at 5:44
• See this. The last numerator should also be a linear polynomial.
– xbh
Apr 4 at 6:21
• Alright, directly, you shall assume the fraction be $$\frac Ax + \frac {Bx + C} {x^2+1} + \frac{Dx + E}{(x^2+1)^2}.$$ The case when repeated linear factors appears cannot apply to your question because the summand in question has a denominator being a power of a quadratic polynomial, not linear, and the numerator cannot just be constants.
– xbh
Apr 4 at 6:46
• Also I should put it out that the part I want you to read is "Over the Reals -- General result" in the link.
– xbh
Apr 4 at 6:51
• Yeah, but… if you don't use the standard form, then you will be getting nowhere in most cases… like right now :(
– xbh
Apr 4 at 6:55