Quintic Equations

Date: 1/27/96 at 11:36:21
From: Anonymous
Subject: definition

I'm taking an Internet course and it's being held in a math lab at UIC 
in Chicago.  On the wall is a poster called "Solving Quintic".  I tried 
looking up the word Quintic in the Webster's on-line dictionary and 
found nothing.  What does Quintic mean?  Is it a math term or a product 
name?  I'm just an elementary school teacher and am curious.

Date: 1/27/96 at 12:26:29
From: Doctor Sarah
Subject: Re: definition

Hello there -

You're on the right track using the Internet to look for the quintic. 
If you have an online dictionary you'll probably also have access to a 
Web browser, right?  You can try using a searcher like Alta Vista at 

    http://altavista.digital.com/   

to search for 'quintic'.  Here's one of the pages it finds:

STEPS TO THE QUINTIC

   http://www.wri.com/posters/quintic/main.html   

The page takes you through the quadratic, cubic, and quartic equations 
on the way to the quintic.  You know what the quadratic equation looks 
like, right?

   ax^2 + bx + c = d (where a, b, c, and d are real numbers and a does 
not equal zero)

The quintic looks like this:

    ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0, x

I'll let you look up the answer (found using Mathematica) on the Web 
page.  Here's some of what it says:

"Root objects are an implicit way to represent the solution. They can be
differentiated and expanded out in series, and with approximate
numerical values for the coefficients, they immediately yield a 
numerical solution. Of course, we can solve a quintic with numerical 
coefficients immediately by using the built-in Mathematica function 
NSolve.

"Ruffini (1799) and Abel (1826) proved that it is not possible to give 
an explicit solution for the general quintic equation with symbolic
coefficients in terms of square roots, cube roots, and so on. Is there 
an explicit solution to the quintic with symbolic coefficients? Yes! In 
the late 1800s, several mathematicians constructed such solutions. 
However, it was necessary to go beyond the extraction of roots and to 
use elliptic and hypergeometric functions. Mathematica can handle these 
higher mathematical functions in the same way as ordinary trigonometric 
or exponential functions. Combined with Mathematica's algebraic
capabilities, this makes it possible to implement various symbolic
solutions to the quintic."

There's a lot more.  Enjoy!

-Doctor Sarah,  The Math Forum