Polynom and decomposition

Question

I need to know I can decompose into simple elements

$$\frac X{(X+1)^4 (X^2 +1)}$$

What is the easiest way?

Answer 1

As there is a pole of high order, the fastest way is division by increasing powers. I'll sketch it.

We first have to make the substitution $X+1=T$, so the pole of order $\color{red}4$ will be $0$. The fraction can be rewritten as $$\frac{X}{X^4(X^2+1)}=\frac{T-1}{T^4(T^2-2T+2)}.$$ Now perform the division by increasing powers of $T-1$ by $2-2T+T^2$ up to order $\color{red}3$. You obtain the equality: $$T-1=\Bigl(-\frac12+\frac14T^2+\frac14T^3\Bigr)(2-2T+T^2)+\frac14T^4(1-T),$$ from which you deduce, dividing by $T^4(2-2T+T^2)$: $$\frac{T-1}{T^4(T^2-2T+2)}=-\frac1{2T^4}+\frac1{4T^2}+\frac1{4T}+\frac{1-T}{2-2T+T^2},$$ and finally, going back to $X$: $$\frac{X}{X^4(X^2+1)}==-\frac1{2(X+1)^4}+\frac1{4(X+1)^2}+\frac1{4(X+1)}-\frac{X}{X^2+1}.$$