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A polynomial equation of degree greater than four will in general have no solution formula. But what are some typical cases one should be aware of as a practical person in which there are solutions?

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    It depends on what you mean by "formula." I want to caution you that the answers below which use the word "solvable" are doing so in a rather precise technical sense of the word (see http://en.wikipedia.org/wiki/Galois_theory#Solvable_groups_and_solution_by_radicals) and it's not clear to me whether you actually care about this notion of solvability.2012-03-23

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Another example that come up fairly often is $a x^{2n} + b x^n + c = 0$

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    Yes, of course. If $f$ is solvable and $g - c$ is solvable for every constant $c$, then $f \circ g$ is solvable.2012-03-24
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The example of greatest practical importance is $x^n=1$. The solutions are the $n$-th roots of unity. They come up moderately often in applied work, and are omnipresent in pure mathematics.

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    More generally, every abelian extension is solvable ;).2012-03-23