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The root of a linear equation is easy to derive. The quadratic formula is not much more difficult. Cubic and quartic formulas get ugly.

Is there a general formula which can be used to generate all of these, and which can give is information about quintic equations and beyond?

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  • $\begingroup$ You might already be aware of the fact that for general polynomials of degree greater than or equal to $5$, the roots cannot be expressed by a formula involving the composition of field operations and extracting $n$-th roots. $\endgroup$ – Tob Ernack Jun 3 '17 at 5:41
  • $\begingroup$ General formula : No ... from a practical point of view, if you just want the value(s) in a specific instance, ask Wolfie ... wolframalpha.com/input/?i=graph+y%3Dx%5E5%2Bx%2B1 ... wolframalpha.com/input/?i=roots+x%5E5+%2Bx%2B1 $\endgroup$ – Donald Splutterwit Jun 3 '17 at 5:55
  • $\begingroup$ If you want to know about rational roots, the rational root test is a nice place to start. $\endgroup$ – Oiler Jun 3 '17 at 5:59
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Victor Protsak at mathoverflow gave this answer:

https://mathoverflow.net/a/32261/25104

which as mathematica program for a degree 6 polynomial is:

(*start*)
Clear[x, k];
n = 6;
a = 1/3;
N[Sum[Binomial[n*k, k]*a^((n - 1)*k + 1)/((n - 1)*k + 1), {k, 0, 
   2000}], 20]
x = N[Sum[
   Binomial[n*k, k]*a^((n - 1)*k + 1)/((n - 1)*k + 1), {k, 0, 3000}], 
  20]
Print["If this is zero then x is solution to the equation:"]
x^n - x + a
(*end*)
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