$$x^6+x^2+1 = 0$$
Answer
$$ \begin{matrix}x_1 = 0.88084+0.65933i & x_2 = 0.88084-0.65933i & x_3 = 0.82603i \\[1 em] x_4 = -0.82603i & x_5 = -0.88084+0.65933i & x_6 = -0.88084-0.65933i \end{matrix} $$
Explanation
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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