<s>[INST] Find remainder of division of $x^3$ by $x^2-x+1$ I am stuck at my exam practice here. 

The remainder of the division of $x^3$ by $x^2-x+1$ is ..... and that of $x^{2007}$ by $x^2-x+1$ is .....

I tried the polynomial remainder theorem but I am not sure if I did it correctly.
By factor theorem definition, provided by Wikipedia,

the remainder of the division of a polynomial $f(x)$ by a linear polynomial  $x-r$ is equal to $f(r)$.

So I attempted to find $r$ by factorizing $x^2-x+1$ first but I got the complex form $x=\frac{1\pm\sqrt{3}i}{2}=r$. 
$f(r)$ is then $(\frac{1+\sqrt{3}i}{2})^3$ or $(\frac{1-\sqrt{3}i}{2})^3$ which do not sound right.
However, the answer key provided is $-1$ for the first question and also $-1$ for the second one. Please help.
 [/INST] Another way if you're familiar with modular arithmetic is to work modulo $x^2-x+1$, in which case we have $x^2\equiv x-1$ and thus
$$x^3\equiv xx^2\equiv x(x-1)\equiv x^2-x\equiv (x-1)-x\equiv -1.$$ This can be extended to your other question by noting that $x^{2007}=\left(x^3\right)^{669}$.
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