Question

For $n\in N,x^{n+1}+(x+1{)}^{2n-1}$ is divisible by

• A. $x$
• B. $x+1$
• C. $x^2+x+1$
• D. $x^2-x+1$

Answer 1

The correct option is C $x^2+x+1$

For $n=1$, we have
$x^{n+1}+(x+1{)}^{2n-1}$
$=x^2+(x+1)=x^2+x+1$, which is divisible by $x^2+x+1.$
For $n=2$, we have
$x^{n+1}+(x+1{)}^{2n-1}$
$=x^3+(x+1{)}^3$
$=(2x+1)(x^2+x+1)$, which is divisible by $x^2+x+1.$
Hence, option (c) is true.