Polynomial ring/X^(n-1)+...+1/X-1/Representation of 1/Exercise

Let ${}n\geq 2$.

1. Perform in ${}\mathbb {Q} [X]$ the division with remainder ${}P$ divided by ${}Q$ for the polynomials ${}P=X^{n-1}+X^{n-2}+\cdots +X^{2}+X+1$ and ${}T=X-1$.
2. Find a representation of ${}1$ as a linear combination of these polynomials.

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