Complicate the expression for $\frac{p-1}{p}x+\frac{a}{p}x^{1-p}-a^{1/p}$

Question

I am interested in finding a way to go between the RHS and LHS of the following equation $$\frac{p-1}{p}x+\frac{a}{p}x^{1-p}-a^{1/p}=(x-a^{1/p})\left[\frac{p-1}{p}-\frac{1}{p}\left(\sum_{n=1}^{p-1}\left(\frac{a^{1/p}}{x}\right)^n\right)\right]$$ In the book, it is said it is an "easily" derived formula, however, even after I reversed engineeres the RHS (recognizing geometric sum and simplifying as much as possible) I have been unable to find a way from the left to the right side.
I feel like I must be missing something since this manipulation was supposed to be easy but I do not see it. How would I have gone from the LHS to the RHS?

Answer 1

Working on another way round usually is not easy.

Observe that

\begin{array}{rcl} X^2-2X+1 &=& (X-1)(X-1) \\ 2X^3-3X+1 &=& (X-1)(2X^2-X-1) \\ 3X^4-4X+1 &=& (X-1)(3X^3-X^2-X-1) \\ 4X^5-5X+1 &=& (X-1)(4X^4-X^3-X^2-X-1) \\ & \vdots & \\ (p-1)X^{p}-pX^{p-1}+1 &=& (X-1)[(p-1)X^{p-1}-X^{p-2}-\ldots-X-1] \\ X &=& \dfrac{x}{a^{1/p}} \\ \dfrac{p-1}{p}x+\dfrac{a}{p}x^{1-p}-a^{1/p} &=& \dfrac{(p-1)X^{p}-pX^{p-1}+1}{p}\cdot \dfrac{a^{1/p}}{X^{p-1}} \\ &=& a^{1/p}(X-1) \left[ \dfrac{p-1}{p}-\dfrac{1}{p} \left( 1+\dfrac{1}{X}+\ldots+\dfrac{1}{X^{p-1}} \right) \right] \end{array}

The result follows.

Further point to be noticed: \begin{align*} nx^{n+1}-(n+1)x^{n}+1 &= (x-1)(nx^{n}-x^{n-1}-\ldots-x-1) \\ &= (x-1)^2[nx^{n-2}+(n-1)x^{n-3}+\ldots+2x+1] \end{align*}