<s>[INST] Why is $(x^{2n+1} - (2n+1)x^{n+1} + (2n+1)x^n - 1)/(x-1)^3$ irreducible? Consider the following polynomial
$$
f_n(x)=x^{2n+1} - (2n+1)x^{n+1} + (2n+1)x^n - 1
$$
Try the first $n$, I find that the $(x-1)^3$ is its factor:
$$
f_1(x)=(x-1)^3,\\
f_2(x)=(x-1)^3 (x^2+3x+1),\\
f_3(x)=(x-1)^3 (x^4+3x^3+6x^2+3x+1),\\
\cdots
$$
and that this is the complete factorization of $f_n(x)$ in $\mathbb{Z}[x]$.
But how can you prove that factoring out $(x-1)^3$ gives the complete factorization over $\mathbb{Z}$? That is, how can you prove that $f_n(x)/(x-1)^3$ is irreducible over $\mathbb{Z}$?
PS: I use Mathematica to test $n\leqslant 1000$, all is correct.
 [/INST] $(x-1)^n$ is a factor of the polynomial $f(x)$ with real coefficients iff
$f(1)=f'(1)=f''(1)=\cdots=f^{(n-1)}(1)=0$.
Here
$$f(1)=1-(2n+1)+(2n+1)-1=0,$$
$$f'(1)=(2n+1)-(n+1)(2n+1)+n(2n+1)=0,$$
$$f''(1)=(2n+1)2n-(n+1)n(2n+1)+n(n-1)(2n+1)=0$$
and
$$f'''(1)=(2n+1)2n(2n-1)-(n+1)n(n-1)(2n+1)+n(n-1)(n-2)(2n+1)
=(2n+1)n(n+1)>0.$$
</s>