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There exists no general formula for quintic equations. Then how does WolframAlpha solve quintic or higher degree polynomial.

Is there a sure way to get values of roots of quintic or higher degree polynomials without brute forcing (especially non integer roots)?

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  • $\begingroup$ Perhaps it relies on a mixture of numerical methods and trial and error? $\endgroup$ Commented Aug 6, 2015 at 16:10
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    $\begingroup$ There is a well-known method for determining which rational quintics are solvable by radicals. Is this the sort of thing you would be interested in? $\endgroup$
    – hardmath
    Commented Aug 6, 2015 at 16:16
  • $\begingroup$ Some quintics are solvable by radicals. You can determine whether a given quintic is one of these using Galois theory. $\endgroup$
    – GEdgar
    Commented Aug 6, 2015 at 16:28
  • $\begingroup$ It is possible to solve general 5th degree equations, but not generally with field operations and radicals only. $\endgroup$
    – ccorn
    Commented Aug 6, 2015 at 17:47
  • $\begingroup$ Wolfram alpha uses approximation to solve quintics $\endgroup$
    – Alexander
    Commented Jun 30, 2019 at 17:37

2 Answers 2

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There exists no general formula for quintic polynomials from using addition, subtraction, multiplication, division, and $n$th roots. But numerically, we are extremely good at finding roots of polynomials very quickly.

If you have a root within a known interval (like you might get by simply evaluating it at a million points and graphing it), then naive applications of the Newton-Raphson method will converge extremely quickly.

As an aside, you mention integer and non-integer roots. The rational root theorem means that it is always easy (for a computer) to find all rational roots of a polynomial. It becomes a finite guessing game. This is true brute force, but it's guaranteed to work perfectly.

WolframALpha first checks against a bank of some factorizations it has stored. Then it checks for rational roots. It is possible that it uses Sturm's Theorem to decide if it is solvable by radicals, but I'm not completely sure. What I do know is that it very frequently resorts to raw numerical computations.

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  • $\begingroup$ I believe it's spelled Newton-Raphson. But otherwise great answer! $\endgroup$ Commented Aug 6, 2015 at 17:15
  • $\begingroup$ Hmm, this probably means I've been mispronouncing his name for years, too. Thank you for letting me know. $\endgroup$
    – davidlowryduda
    Commented Aug 6, 2015 at 17:17
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Is there a sure way to get values of roots of quintic or higher degree polynomials without brute forcing $($especially non integer roots$)$ ?

Some quintics $($both reducible and irreducible$)$ are solvable.

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