Question
$$x^5+x^4+x^3+x^2+x+\frac{1}{x}-1 = 0$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} x^5+x^4+x^3+x^2+x+\frac{1}{x}-1 &= 0&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]xx^5+xx^4+xx^3+xx^2+xx+x\cdot\frac{1}{x}-x\cdot1 &= x\cdot0&& \text{cancel out the denominators} \\[1 em]x^6+x^5+x^4+x^3+x^2+1-x &= 0&& \text{simplify left side} \\[1 em]x^6+x^5+x^4+x^3+x^2-x+1 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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