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**

`> `