How to factorize $n^5+n+1$? How to factorize $n^5+n+1$ ?
I think I should break $a^5$ and use a factorization formula. but how is it done?

By trial and error, knowing that there must be a simple solution.


*

*Factoring $n^5+1$ leads nowhere.
$$n^5+n+1=(n+1)(n^4-n^3+n^2-n+1)+n.$$

*Factoring $n^5+n$ leads nowhere.
$$n^5+n+1=n(n+1)(n^3-n^2+n-1)+1.$$

*Factoring $n^5+n^2$ with an artifice fails, but shows some hope
$$n^5+n^2-n^2+n+1=n^2(n+1)(n^2-n+1)-n^2+n+1.$$

*Factoring $n^5-n^2$ instead works !
$$n^5-n^2+n^2+n+1=n^2(n-1)(n^2+n+1)+n^2+n+1.$$

