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From reading a few accounts of the unsolvability of the quintic, I am told that, e.g., there were public contests in which people competed to solve polynomial equations, and that over the course of 200-300 years so much effort went into trying to find a solution for the quintic that people started to doubt that a solution even existed (which did eventually lead to such a result).

Generally, I get the impression that a lot of brainpower went into trying to solve polynomial equations (both trying to come up with general solution formulas and solving particular instances) and that people were really interested in this problem.

Why? What was so interesting about solving polynomial equations that people spent so much effort on the problem?

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The conics as geometric shapes were investigated comprehensively by many ancient Greek mathematicians. After Descartes's innovation of introducing coordinates, they were seen as essential examples of 2nd order polynomial equations. The first order equation was simply the straight line and so simple that no-one bothered investigating its geometric properties.

This covers first and second order equations. It's natural to then investigate higher order equations and also to investigate how to solve them. After all, that's one of the points of an equation. Algebraic methods were developed for the third and then the fourth and there they stalled until Galois showed the fifth order equation didn't have a general solution that could be written down as a formula using the algebraic operators (addition & multiplication) and the taking of $n$th roots.

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