I posted this question a few days ago but the images didn’t work so nobody knew what I was talking about.
I'm self-studying through Amann/Escher Analysis 1 and I'm stuck on a problem in I.8. Here's the problem:
It seems clear that the goal should be to show that $X_m$ is a zero of the right hand side. From there, I can use the linear factor theorem for polynomials in 1 indeterminate and the fact that the right hand side has at most $m$ factors.
I can see that whenever I evaluate the function at $X_i$ for $1 \leq i \leq m$, each element in each term pairs off nicely with an element in another term. This is the last part of an exercise in symmetric polynomials, so I think it has something to do with that, but I'm having trouble putting it all together. I also don't know very much about symmetric polynomials. Here is the rest of the exercise, which I have completed, and which constitutes everything I am "allowed to know" about symmetric polynomials for the purposes of this problem.
Normally I would just ask for a hint but I'm sick of this problem. I've been working on it for a while.
I tried to do an induction today and I found that given the induction hypothesis, the right hand side is equal to $(-1-X_m)(X-X_1)…(X-X_{m-1})+X^{m}$ but I am not confident in this calculation.