The question is this -
If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $h(x)=xf(x^3)+x^2g(x^6)$ is divisible by $x^2+x+1$, then which of the following are true?
1. $f(1)=g(1)$
2. $f(1)=-g(1)$
3. $h(1)=0$
So I wrote out this $$xf(x^3)+x^2g(x^6)=Q(x).(x^2+x+1)$$ where $Q(x)$ is the quotient polynomial.
Then I tried substituting a few values like $1$ and $0$, but these get me nowhere. The $Q(x)$ is making problems and also the fact that $x^2+x+1=0$ does not have easy to substitute roots.
Can anyone give any hints or general tips regarding questions like these? Thank you.
Edit Real coefficients to the polynomials might have to be assumed, I'm not sure.
Let $\omega=-\frac12+\frac i2\sqrt 3$ be a root of $x^2+x+1$. Then $\omega^3=1$ and $\omega^2=\overline \omega$ is the other root. Therefore, $$\omega f(1)+\overline\omega g(1) =Q(\omega)\cdot 0 $$ and also $$\overline \omega f(1)+\omega g(1) =Q(\omega^2)\cdot 0 $$ from which we conclude $f(1)=g(1)=0$, so that all three claims are correct.