Question

If $f\left(x\right)$ and $g\left(x\right)$ are two polynomial such that the polynomial $h\left(x\right)=xf\left(x^3\right)+x^{2}g\left(x^6\right)$ is divisible by $x^2+x+1,$ then

• A. $f\left(1\right)=g\left(1\right)$
• B. $f\left(1\right)=-g\left(1\right)$
• C. $h\left(1\right)=0$
• D. $h(-1)=0$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $h\left(1\right)=0$
B $f\left(1\right)=g\left(1\right)$
C $f\left(1\right)=-g\left(1\right)$

The roots of $x^2+x+1=0$ are $w$ and $w^2$

So, $h\left(w\right)=0$ and $h\left(w^2\right)=0$

$\Rightarrow wf\left(1\right)+w^{2}g\left(1\right)=0$ and $w^{2}f\left(1\right)+wg\left(1\right)=0$

$\Rightarrow f\left(1\right)=g\left(1\right)=0$

$\therefore h\left(1\right)=f\left(1\right)+g\left(1\right)=0.$