Find the root of the polynomial : $P(X) = 1+2X+3X^2+...+(n-1)X^{n-2}+nX^{n-1}+(n-1)X^n+...+3X^{2n-4}+2X^{2n-3}+X^{2n-2}$

Question

During an exam I got the following exercise :

Let $n \geq 3$, find all the roots of the following polynomial :

$$P(X) = 1+2X+3X^2+...+(n-1)X^{n-2}+nX^{n-1}+(n-1)X^n+...+3X^{2n-4}+2X^{2n-3}+X^{2n-2}$$

It’s hard to show any real attempt since I don’t know how to proceed. Obviously I tried some factorisation when $n$ is small but didn’t find anything. Nevertheless there is for sure a link with the root of unity (I can’t explain while but I feel it).

Answer 1

$$1+2x+3x^2+\cdots+nx^{n-1}+(n-1)x^{n-2}+\cdots+2x^{2n-3}+x^{2n-2} =(1+x+\cdots +x^{n-1})^2.$$

Can you solve $$1+x+\cdots+x^{n-1}=0?$$

Answer 2

Let $X=10$. What are the factors of $121$ and $12321$?