Is it possible factor out $(x - b)$ from $(x^n-b^n)$ when $n$ is a fraction? Factoring out $(x-b)$
But how to factor out $(x-1)$ when the polynomial has the power of a negative integer where $n=-4$
$x^{-4}-1$
And even worst how to perform the factorization when the polynomial is to the power of an fraction? $n=-4/7$
$x^{-4/7}-1$

For your example :
You could call $x^{-1/7}=\frac{1}{x^{1/7}}=y$
So
$$y^4-1=(y-1)(y+1)(y^2+1)=\left(\frac{1}{x^{1/7}}-1\right)(y^3+y^2+y+1)$$
$$=-\frac{1}{x^{1/7}}\left(x^{1/7}-1\right)(y^3+y^2+y+1)$$
