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Find the quotient of polynomials $f(x)=x^{2n}-nx^{n+1}+nx^{n-1}-1$ and $g(x)=x-1$.

I know that by the remainder theorem the remainder is $f(1)=1-n+n-1=0$ so we can write $f(x)=(x-1)q(x)$ but beyond that I don't know what to do even though I tried doing the division, as well as guessing the quotient to find its coefficients using the polynomial equality theorem, but these approaches got me nowhere.

Aaron Hendrickson
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user562834
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    You might consider factoring $x^{2n}-1$ and $-nx^{n-1}(x^2-1)$ separately then adding the results together... – abiessu Jun 21 '21 at 17:34
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    Write $f(x)$ as $x^{2n}-1-nx^{n-1}(x^{2}-1)$ and use the identity $a^m-b^m=(a-b)(a^{m-1}+a^{m-2}b+...+ab^{m-2}+b^{m-1})$ with $a=x$, $b=1$ and $m=2n$ and with $m=2$. – plop Jun 21 '21 at 17:35
  • Did it, got $(x^{n-1}+x^{n-2}+\dots+1)(x^n+1)-nx^{n-1}(x+1)$. – user562834 Jun 21 '21 at 17:40
  • @user562834 You will get $(x^{n-1}+x^{n-2}+\dots+1)(x+1)-nx^{n-1}(x+1)(x-1)$ and when you divide by $x+1$ you will get $(x^{n-1}+x^{n-2}+\dots+1)-nx^{n-1}(x-1)$ – Asher2211 Jun 21 '21 at 17:57

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