Find remainder when $F(x)$ is divided by $P(x)$ Find the remainder when $F(x)=x^{276}+12$ is divided by $P(x)=x^2+x+1$?

You have $F(x) = (x^2+x+1) q(x) + r(x) = \frac{x^3-1}{x-1} q(x) + r(x)$, with $\partial r <2$. Let $x$ be a cube root of $1$ different from $1$, then $F(x) = r(x)$, and since $276 = 0 \mod 3$, we have $x^{276} = 1$, and so $F(x) = 13$ at both 'non one' roots.
Hence $r(x) = 13$.
