Is there a way to find all roots of a polynomial equation?

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Is there a way to find all roots of a polynomial equation?

Lets say $$x^5+ax^4+bx^3+cx^2+dx+e=0$$

how to find its all roots?

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2 Answers

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A theorem by Abel and Ruffini (see e.g. ) states that there is no general way of expressing (explicitly) the roots of a polynomial of order 5 or more or said differently, that there exists polynomials of order 5 or more for which it is impossible to do so. (However, and as was mentioned in other comments, there exists ways of approximating the roots with arbitrary precisions)

Now there exists methods to find the number of roots in a particular zone. See e.g. Sturm method, Budan-Fourier, Routh-Hurwitz (argument principle),

and also, exclusion/inclusion theorem (e.g. van der Sluis' theorem, Laguerre's theorem).

So the brief answer to your question is: no there is not. However there are a lot of methods to characterize the localization of the roots or to approximate the roots.

EDIT: precisions added following comments by Franklin.vp.

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I'm reading this paper of AMS really, an equation of degree 5, is not easy to solve, unless that you have some properties over the coefficients.

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