Question

$(1+x{)}^n-nx-1$ is divisible (where $nϵN)$

• A. by $2x$
• B. by $x^2$
• C. by $2x^3$
• D. all of these

Answer 1

The correct option is B by $x^2$

$(1+x{)}^n-nx-1$
$=[1+nx+{}^n{C}_2{x}^2+{}^n{C}_3{x}^3+...{}^n{C}_n{x}^n]-nx-1$
$={}^n{C}_2{x}^2+{}^n{C}_3{x}^3+...{}^n{C}_n{x}^n$
Hence the above expression is divisible by $x^2$.