<s>[INST] Finding roots of complex equation? Determine all roots of the equation $x^6+(3+i)x^3 + 3i = 0$ in $\mathbb{C}$ Express answers in standard form
 [/INST] Let $x^3 = z$. We then have that
$$z^2 + (3+i)z + 3i = 0 \implies (z+3)(z+i) = 0 \implies z = -3, -i \implies x^3 = -3,-i$$
Hence,
$$x = \sqrt[3]{-3},\sqrt[3]{-3} \omega, \sqrt[3]{-3} \omega^2, i, i\omega, i\omega^2$$where $\omega$ is the complex cube-root of $1$ and $\sqrt[3]{-3}$ is the real number $y$ such that $y^3 = -3$.
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