If f(x) and g(x) are two polynomials such that the polynomial P(x)=f(x^3)+xg(x^3) is divis…

Question

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

Answer: 0

Accepted Answer

Since $\omega,\omega^2$ are roots of $x^2+x+1=0$, $P(\omega)=P(\omega^2)=0$.

$P(\omega)=f(1)+\omega g(1)=0$ and $P(\omega^2)=f(1)+\omega^2g(1)=0$. Subtracting gives $(\omega-\omega^2)g(1)=0 \implies g(1)=0$, then $f(1)=0$.

$P(1)=f(1)+g(1)=0.$

Solutions · 1