Cubic Equations

Date: Thu, 08 Dec 1994 09:31:51 -0500 (EST) 
From: Anonymous
Subject: Cubic Equations

Dear Dr. Math,

In calculus, we were looking for the solutions for a third degree polynomial
equation.  Using my TI-85 to find the solutions, I stumbled upon what appears
to be an interesting observation. 

Given a cubic on the form ax^3 + bx + c, I have conjectured that the absolute
value of the sum of any two zeros is equal to the absolute value of the third. 
 I have, so far, been unable to give a general proof.  Do you have any
suggestions? 

Tom Vitolo '96
Kent School
Kent, CT  06757

please reply c/o Tom Roney
Kent School
roneyt@delphi.com

Date: Thu, 8 Dec 1994 11:52:23 -0500 (EST)
From: Dr. Ken
Subject: Re: Cubic Equations

Hello there!

That's a very interesting observation you've made.  Here's what I would
suggest you look at:

If a degree three polynomial has three real solutions, then it can be
factored into three degree one polynomials.  So we can write the polynomial
in the form r(x-a)(x-b)(x-c).  When you multiply this out and put it in
standard form, what do you get for the coefficients on x^3, x^2, x, and the
constant term?

More specifically, if the coefficient of x^2 is zero, what relationship must
hold between a, b, and c?  I hope this will give you some insight into your
solution.

-Ken "Dr." Math