$$7a^7+4a-1 = 0$$
Answer
$$ \begin{matrix}a_1 = 0.24989 & a_2 = -0.82556+0.45776i & a_3 = -0.82556-0.45776i \\[1 em] a_4 = -0.04017+0.91719i & a_5 = -0.04017-0.91719i & a_6 = 0.74079+0.46084i \\[1 em] a_7 = 0.74079-0.46084i & \\[1 em] \end{matrix} $$
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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