Question

When the polynomial $5x^3+Mx+N$ is divided by $x^2+x+1$, the remainder is 0. Then find the value of $M+N$.

• A. $M+N=3$
• B. $M+N=-3$
• C. $M+N=5$
• D. $M+N=-5$

Answer 1

Answer: (D)

$M+N=-5$

Let $f\left(x\right)=5x^3+Mx+N$.
Also, $x^2+x+1=(x-\omega )(x-{\omega }^2)$
$f\left(x\right)$ is divisible by $x^2+x+1$. Hence,
$f\left(\omega \right)=5+M\omega +N=0$
and $f\left({\omega }^2\right)=5+M{\omega }^2+N=0$
$⟹M=0;N=-5$
$⟹M+N=-5$
Ans: D