Question
$$x+\frac{1}{x^4} = 5$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} x+\frac{1}{x^4} &= 5&& \text{multiply ALL terms by } \color{blue}{ x^4 }. \\[1 em]x^4x+x^4\cdot\frac{1}{x^4} &= x^4\cdot5&& \text{cancel out the denominators} \\[1 em]x^5+1 &= 5x^4&& \text{move all terms to the left hand side } \\[1 em]x^5+1-5x^4 &= 0&& \text{simplify left side} \\[1 em]x^5-5x^4+1 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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