<s>[INST] Prove that 1 is a triple root of a polynomial I'm studying for an exam and trying to prove whether 1 is a triple root for the polynomial:
$$x^{2n+1}-(2n+1)x^{n+1}+(2n+1)x^n-1$$
for every $n\geq1$.
In our math class we never solved such a problem. So far we only used horner's scheme to prove that someone is a root, double root or triple root.
Can you please help me solve this problem?
 [/INST] Let $f_n(x)=x^{2n+1}-(2n+1)x^{n+1}+(2n+1)x^n-1$. We will prove the statement by induction on $n$.
For $n=1$ we get $f_1(x)=x^3-3x^2+3x-1=(x-1)^3$ and the statement holds.
Now assume $f_n(x)$ has a factor $(x-1)^3$, then take
\begin{align}
f_{n+1}(x)-xf_n(x)
&=x^{2n+3}-x^{2n+2}-2x^{n+2}+2x^{n+1}-1+x\\
&=x^{2n+2}(x-1)-2x^{n+1}(x-1)+(x-1)\\
&=(x-1)(x^{2n+2}-2x^{n+1}+1)\\
&=(x-1)(x^{n+1}-1)^2\\
&=(x-1)^3\left(\frac{x^{n+1}-1}{x-1}\right)^2\\
&=(x-1)^3\left(1+x+x^2+\dots+x^n\right)^2.
\end{align}
The RHS has a factor $(x-1)^3$, so does $f_n(x)$ (by the induction hypothesis), hence $f_{n+1}(x)$ has a factor $(x-1)^3$.
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