$$x^8+x^4 = 1$$
Answer
$$ \begin{matrix}x_1 = 0.88665 & x_2 = -0.88665 & x_3 = 0.88665i \\[1 em] x_4 = -0.88665i & x_5 = -0.7975+0.7975i & x_6 = -0.7975-0.7975i \\[1 em] x_7 = 0.7975+0.7975i & x_8 = 0.7975-0.7975i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x^8+x^4 &= 1&& \text{move all terms to the left hand side } \\[1 em]x^8+x^4-1 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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