Is it possible factor out $(x - b)$ from $(x^n-b^n)$ when $n$ is a fraction?

Question

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? $n=-4/7$

$x^{-4/7}-1$

Answer 1

For your example :

You could call $x^{-1/7}=\frac{1}{x^{1/7}}=y$

So $$y^4-1=(y-1)(y+1)(y^2+1)=\left(\frac{1}{x^{1/7}}-1\right)(y^3+y^2+y+1)$$ $$=-\frac{1}{x^{1/7}}\left(x^{1/7}-1\right)(y^3+y^2+y+1)$$

Answer 2

To answer the first part of your question: $x^{-n}-1$ can be rewritten as $(-1/x^n)*(x^n-1)$, so you just factor $x-1$ out of the $x^n-1$ and leave the $-1/x^n$ on.

$$x^{-4}-1=(-1/x^4)*(x^4-1)$$

$$(-1/x^4)*(x^4-1)=((-1/x^4)*(x^2+1)*(x+1))*(x-1)$$