Question
$$500s^6+12000s^5+100525s^4+372950s^3+\frac{1286475}{2}s^2+620050 = 0$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} 500s^6+12000s^5+100525s^4+372950s^3+\frac{1286475}{2}s^2+620050 &= 0&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2\cdot500s^6+2\cdot12000s^5+2\cdot100525s^4+2\cdot372950s^3+2 \cdot \frac{1286475}{2}s^2+2\cdot620050 &= 2\cdot0&& \text{cancel out the denominators} \\[1 em]1000s^6+24000s^5+201050s^4+745900s^3+1286475s^2+1240100 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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Polynomial Equations Solver