$$x^5-x^4+\frac{1}{10} = 0$$
Answer
$$ \begin{matrix}x_1 = -0.5075 & x_2 = -0.06262+0.5348i & x_3 = -0.06262-0.5348i \\[1 em] x_4 = 0.81637+0.1147i & x_5 = 0.81637-0.1147i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x^5-x^4+\frac{1}{10} &= 0&& \text{multiply ALL terms by } \color{blue}{ 10 }. \\[1 em]10x^5-10x^4+10\cdot\frac{1}{10} &= 10\cdot0&& \text{cancel out the denominators} \\[1 em]10x^5-10x^4+1 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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