Question

If $p,q,r$ are positive integers, then remainder when
$x^{3p}+x^{3q+1}+x^{3r+2}$
is divided by $x^2+x+1$ is

• A. $0$
• B. $1$
• C. $x+1$
• D. $x-1$

Answer 1

The correct option is A $0$

Let $f\left(x\right)=x^{3p}+x^{3q+1}+x^{3r+2}$ and $x^2+x+1=(x-{\omega }^2)(x-\omega )$
Thus, dividing by $x^2+x+1$ is equivalent to dividing the number by $\omega$ and ${\omega }^2$
So, by remainder theorem, remainder when the divisor is $\omega =f\left(\omega \right)=1+\omega +{\omega }^2$ since ${\omega }^3=1$
Similarly, $f\left({\omega }^2\right)=0$ and thus, the remainder has to be 0 when divided by $x^2+x+1.$