Cubic EquationsDate: 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 |
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