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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).

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

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

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  • $\begingroup$ Yes I can. But how did you find this factorization ? $\endgroup$ – Interesting problems Nov 4 '18 at 8:30
  • $\begingroup$ @Interestingproblems Hint: multiply your original polynomial by $(1-x)^2$. The binomial expansion of $(1-x)^{-2}$ suggests this is a good idea. $\endgroup$ – J.G. Nov 4 '18 at 12:39
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Let $X=10$. What are the factors of $121$ and $12321$?

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