<s>[INST] Remainder of a Polynomial when divided by a polynomial with nonreal roots What will be the remainder when $x^{2015}+x^{2016}$ is divided by $x^2+x+1$ without using the fact that $x^2+x+1$ has roots as non real cube roots of unity.
 [/INST] Hint:  using that $a^3-1=(a-1)(a^2+a+1)$ and so $\,x^{3k}-1=(x-1)(x^2+x+1)(\ldots)\,$:
$$
\big(x^{2016}\color{red}{-1}\big)+\big(x^{2015}\color{red}{-x^2}\big)\color{red}{+1+x^2} \\
 =\left(\left(x^3\right)^{672}-1\right)+x^2\left(\left(x^3\right)^{671}-1\right)+\big(x^2\color{blue}{+ x} + 1\big)\color{blue}{- x}
$$
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