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Are there any specific conditions for coefficients of a polynomial of degree 6 to Solve or find at least one of its roots? I do not need numerical method for solving. I want some analytical methods to find the roots if they exists. The polynomial is here:

$$ \ x^{6} + 6 \Delta x^{5} + (4 \Delta ^{2} -36) x^{4} + (-24 \Delta ^{2} -96 \Delta ) x^{3}+(192+112 \Delta ^{2}-32 \Delta ^{4} ) x^{2} +(256 \Delta ^{2} +256 \Delta ) x-512 \Delta ^{2} -256=0\,. $$

$\Delta$ is an Anisotropic quantity between 0 and 10.

illia
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    what is $$\Delta$$? – Dr. Sonnhard Graubner Jan 09 '17 at 14:04
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    If $\Delta=\pm i/\sqrt 2$, I can tell you that $x=0$ is a solution ... – Hagen von Eitzen Jan 09 '17 at 14:10
  • Where did the polynomial come from? –  Jan 09 '17 at 14:14
  • The question makes no sense unless you specify what do you mean by "analytical solution" and what is $\Delta$ precisely. For example, if $\Delta$ is not expressible in radicals, then the roots will (most likely) not be expressible in radicals either. Overall, whether or not the roots can be expressed in radicals depends on the value of $\Delta$. If you allow special functions, then I believe it is possible to express the roots of the general sextic equation analytically. – Yuriy S Jan 09 '17 at 14:18
  • Assuming $\Delta$ real, since $\lim_{x\to\pm\infty}P(x)=+\infty$ and $P(0)<0$, there exist (at least) a couple of roots. – Miguel Jan 09 '17 at 14:21
  • $\Delta$ is an Anisotropic quantity between 0 and 10. – illia Jan 09 '17 at 18:47

2 Answers2

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Generally speaking, for nearly every polynomial of degree greater than 4, you cannot express the results in terms of arithmetic and roots — you need to use some sort of special function; e.g. the quintic can be solved with the Bring radical

But if you're going to use special functions anyways, you might as well use the special function "root of this polynomial", since it is a fairly simple special function and has an extremely simple relationship to the roots of your polynomial

Among the rare exceptions, you can identify some (maybe all) of them by:

  • Using a factoring algorithm.
  • Computing the Galois group of the polynomial over its coefficient field.

WolframAlpha doesn't (immediately) find a simpler form for the roots; thus I'm inclined to expect that this isn't one of the exceptions.

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As described in the links provided in this post, sixth degree polynomials have roots analytically solvable using the Kampé de Fériet functions.

(Though it is horrendous IMO)

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