How do I use the partial fractions technique in this case?

Question

How do I use the partial fractions technique in this case?

$$\frac{(x - 1)}{(x^2 - x + 1)(x + 1)}$$

Answer 1

The partial fraction decomposition takes the form $$F(x)=\frac{x-1}{(x^2-x+1)(x+1)}=\frac{a}{x+1}+\frac{bx+c}{x^2-x+1}$$

and notice that $$a= F(x)(x+1)\bigg|_{x=-1}=-\frac23$$ and $$0=\lim_{x\to\infty}xF(x)=a+b\Rightarrow b=\frac23$$ and finaly $$F(0)=-1=a+c\Rightarrow c=-\frac13$$