For what $A$ and $B$ does the polynomial $A x^{n+1}+B x^{n}+1$ have the number $x=1$ as a root of multiplicity at least two?
$f(1)=A+B+1, f^{\prime}(1)=(n+1) A+n B$. Solving the system $A+B+1=(n+1) A+n B=0$, we find $A=n, B=-(n+$ $1)$.

## Answer

$A=n, B=-(n+1)$.