Explicit formula for a root of a degree 5 polynom $ax^5 + b x^4 + c x^3 + d x^2 + e x + f$

Asked Jul 28 '19 at 21:50

Active Jul 29 '19 at 06:30

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Galois theory shows that the root of polynoms of degree 5 and more cannot be computed explicitly using radicals except for some specific polynoms. If we do not restrict the explicit formula to use radicals and use also hypergeometric functions we do have an explicit formula ( How to solve fifth-degree equations by elliptic functions? closed-form expression for roots of a polynomial How do you solve 5th degree polynomials?)

Can you provide the fully explicit expression ?

The close-form should be self-sufficient and completely explicit :

Only one equal sign : $x= ...$
No reference to another function except the usual operations, trigonometric and log functions
Use of integrals and series ($\Sigma$) are allowed (to explicit the hypergeometric functions).

edited Jul 29 '19 at 06:30

DDS

asked Jul 28 '19 at 21:50

Florian

2

Why exactly do you want such an expression? – Peter Foreman Jul 28 '19 at 21:58
3

See Bring radical, particularly its series. – lhf Jul 28 '19 at 22:02
@Peter, I keep hearing that "polynoms of degree 5 cannot be solved". I expect that such expression can [durably] impress the students's mind. – Florian Jul 29 '19 at 05:32
If you allow series, then expressing the roots might be possible using Lagrange's inversion formula. – Tob Ernack Jul 29 '19 at 05:57

0 Answers0