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Is there any way to calculate all zeroes of a $n$-th degree polynomial, like some general formula?

E. Joseph
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user411765
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    No general formula for degree five or more. Can be proved using Galois Theory. – codetalker Jan 31 '17 at 10:48
  • There exists general formulas up to and including $4$-th degree. However some "simple" equations of higher degree can still be solved. For example $x^{100} -1 = 0$. – Zubzub Jan 31 '17 at 11:00
  • @Zubzub I know of crazy closed forms for up to the sixth degree polynomials. – Simply Beautiful Art Jan 31 '17 at 12:44
  • @SimplyBeautifulArt Can you please link it to me ? Not that I don't believe you, just wanna know :) – Zubzub Jan 31 '17 at 13:14
  • @Zubzub I warn you it's not pretty: http://math.stackexchange.com/questions/1397297/how-to-solve-the-general-sextic-equation-with-kamp%c3%a9-de-f%c3%a9riet-functions – Simply Beautiful Art Jan 31 '17 at 13:16
  • @Zubzub and the fifth degree: http://math.stackexchange.com/questions/542108/how-to-transform-a-general-higher-degree-five-or-higher-equation-to-normal-form – Simply Beautiful Art Jan 31 '17 at 13:19
  • @SimplyBeautifulArt Of course if you accept that hypergeometric and elliptic functions are closed form this is true :-P Thank you for the reference ! – Zubzub Jan 31 '17 at 13:29

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No "formula" (multiplication, addition, radicals, etc.) exists for degree $n \geq 5$, as shown by Abel–Ruffini theorem. The result is somewhat technical of nature, and uses Galois theory.

Even more is true: No general algorithm exists for polynomials in $\mathbb N[x]$ (and $\mathbb Q[n]$), as shown in Matiyasevich's theorem.

  • This is very misleading. The Abel-Ruffini theorem only rules out formulas of a certain type, and Matiyasevich's theorem is about integer roots of polynomials in multiple variables. – Eric Wofsey Jan 31 '17 at 10:59
  • You're right. I've updated the answer to specify those. I was simply assuming that OP meant polynomials over $\mathbb Q$. –  Jan 31 '17 at 11:06
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Like @M. Fisher said, there is not general formula for the roots of a polynomial of degree greater or equal than five. Though it is quite hard to prove and it uses Galois' theory.

But you can still find an approximation of any root you want with a given precision using Newton-Raphson's method.

You can find infos here.

E. Joseph
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  • Note that Newton's method will only guarantee to work if the polynomial has degree 2 or less. Otherwise, it might fail, as the search space isn't convex. –  Jan 31 '17 at 10:58
  • @M.Fischer If you are looking for the roots of polynomials in $\mathbb R[X]$, and you initialise the method close enough to a root, there is no problem and it will work fine. – E. Joseph Jan 31 '17 at 11:00
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    The issue here is that you might not know where the root is. –  Jan 31 '17 at 11:02