Question

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P(x)=f\left(x^3\right)+xg\left(x^3\right)$ is divisible by $x^2+x+1$, then $P(1)$ is equal to________.

Answer 1

Answer: 0

$P(x)=f\left(x^3\right)+xg\left(x^3\right)$
$P(1)=f(1)+g(1)\dots$(1)
Now $P(x)$ is divisible by $x^2+x+1$
$\Rightarrow P(x)=Q(x)\left(x^2+x+1\right)$
$P(w)=0=P\left(w^2\right)$ where $w,w^2$ are non-real cube
roots of units
$P(x)=f\left(x^3\right)+xg\left(x^3\right)$
$P(w)=f\left(w^3\right)+wg\left(w^3\right)=0$
$f(1)+\mathrm{wg}(1)=2\dots (2)$
$P\left(w^2\right)=f\left(w^6\right)+w^{2}g\left(w^6\right)=0$
$f(1)+w^{2}g(1)=0\dots (3)$
$(2)+(3)$
$\Rightarrow 2f(1)+\left(w+w^2\right)g(1)=0$
$2f(1)=g(1)\dots (4)$
$(2)-(3)$
$\Rightarrow \left(w-w^2\right)g(1)=0$
$g(1)=0=f(1)$ from $(4)$
from (1) $P(1)=f(1)+g(1)=0$