Dr. Sarah's Maple Demo on Non-Solvability of Quintic Equations by Radicals

The Quadratic, Cubic and Quartic solution formulae are built up from the coefficients of the equation by repeated addition, subraction, multiplication, division and taking roots . Expressions of this kind are called radical expressions.

The following quintic polynomial is not solvable by radicals ;

> f:=unapply(x^5-2*x^3-8*x-2, x);

It does certainly have roots that we can find, but we cannot express these roots only in terms of the combinations of the coefficients of the polynomial . So, this quintic is not solvable by radicals.

Let's see if Maple will solve for the roots for us:

> solve(f(x)=0);

Let's plot this polynomial and look at the roots:

> plot(x^5-2*x^3-8*x-2,x=-2..2.2);

Play around with the domain of the funcion to try and identify the roots, and check to see whether you can find any others. Explain your exploration:

Continued Examination of the Roots of f(x)=x^5-2*x^3-8*x-2, a Quintic Not Solvable by Radicals.


A Quintic Which is Solvable By Radicals