how to do partial fraction decomposition on this equation

Question

I have a fractional function $\frac{1}{x(1+x^{n-1})}$. using PFD: $\frac{A}{(1+x^{n-1})}+ \frac{B}{x}$, that means $Ax+(1+x^{n-1})B=1$.

For this to hold, we need $A=0, B=0, B=1$, which is of course impossible.

Does that mean that this fraction cannot be decomposed? I remember reading that all fractions of polynomials can be decomposed.

Well you need instead of $B$ to have a polynomial with degree $x^{n-2}$ or to simplify $1+x^{n-1}$ — kingW3, Apr 17 '17 at 13:13
Do you mean $A$ instead of $B$? setting $A=-x^{n-2}$ gives $\frac{-x^{n-2}}{1+ x^{n-1}}$. I'm not sure how to integrate that. — user56834, Apr 17 '17 at 13:35

score 1 · Answer 1 · answered Apr 17 '17 at 13:33

1

A and B can be polynomials.

In this case, $A=-x^{n-2},B=1$ Works.

answered Apr 17 '17 at 13:33

marty cohen

Of course, I didn't realize that. That gives $\frac{-x^{n-2}}{1+ x^{n-1}}$. I'm not sure how to integrate that, though? – user56834 Apr 17 '17 at 13:36

1 Answers1