<s>[INST] Find the remainder of the division of polynomials $x^{2007}$ divided by $x^2-x+1$.
I consider to solve this problem, should I break the $x^{2007}$ to find the formula $x^2-x+1$? 
 [/INST] Write: $$\boxed{x^{2007} = k(x)(x^2-x+1)+r(x)}$$ where $r(x)$ is a linear polynomial.
Say $a$ is zero of $x^2-x+1$, then $$ a^2-a+1=0\;\;\;/\cdot (a+1)$$ we get $$a^3+1 =0 \;\;\Longrightarrow \;\;a^3 = -1$$ and if we put $x=a$ in boxed equation we get $$-1= a^{2007} = k(a)\cdot 0+r(a)$$
and the same for other zero $b =\overline{a}$.
So if $r(x) = kx+n$, then we have a system $$-1=ka+n$$ $$-1=kb+n$$
which can easly be solved. 
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