3
$\begingroup$

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?

$\endgroup$
11
  • 1
    $\begingroup$ You assumed the numerator in the middle summand to be a linear polynomial, so why the last one term only has a constant numerator? $\endgroup$
    – xbh
    Apr 4 at 5:44
  • 1
    $\begingroup$ See this. The last numerator should also be a linear polynomial. $\endgroup$
    – xbh
    Apr 4 at 6:21
  • 4
    $\begingroup$ 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. $\endgroup$
    – xbh
    Apr 4 at 6:46
  • 1
    $\begingroup$ Also I should put it out that the part I want you to read is "Over the Reals -- General result" in the link. $\endgroup$
    – xbh
    Apr 4 at 6:51
  • 1
    $\begingroup$ Yeah, but… if you don't use the standard form, then you will be getting nowhere in most cases… like right now :( $\endgroup$
    – xbh
    Apr 4 at 6:55
0
$\begingroup$

If you don't use the standard form, then you will be getting nowhere in most cases… like right now :(

The most general case is the following one [screenshot of the section "Over the reals---General Result" in this link]:

General case for partial fractions

$\endgroup$
3
  • $\begingroup$ So, now we are copying Wikipedia screenshots to StackExchange... $\endgroup$ Apr 4 at 7:42
  • $\begingroup$ Is this deed highly controversial? I shouldn't do this? 😳 $\endgroup$
    – xbh
    Apr 4 at 7:58
  • $\begingroup$ I am not an authority here, wait for possible comments of others. - I would not do it personally. $\endgroup$ Apr 4 at 8:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.