Signs of Roots of 6th-Degree Polynomials
Date: 12/12/2000 at 08:35:44 From: Christopher Subject: Descartes Rule of Signs I've been having some problems with Descartes' Rule of Signs. Can you explain to me why, in a 6th-degree polynomial, the roots can't have two real positive roots, two real negative roots, and two imaginary roots? My teacher told me it's related to the sum and product of the roots. We were told that the maximum number of positive roots is three, and the maximum number of negative roots is three. This is what the possibilities look like, according to her. Real (+) (-) Imaginary 3 3 0 1 3 2 3 1 2 If there are more I really don't know. Regards Chris
Date: 12/12/2000 at 10:51:53 From: Doctor Rob Subject: Re: Descartes Rule of Signs Thanks for writing to Ask Dr. Math, Chris. There are polynomials of degree 6 with real number coefficients that have two positive real roots, two negative real roots, and two complex roots. One such polynomial is: x^6 - 4*x^4 - x^2 + 4. The roots are 1, 2, -1, -2, i, and -i. That tells me that your teacher is referring to a specific polynomial of degree 6 that has a different pattern of signs in its coefficients. Perhaps it was a polynomial like: f(x) = x^6 - 3*x^5 - 7*x^4 + 6*x^3 + 2*x^2 - x - 12 This polynomial has three sign changes in the sequence of its coefficients: + - - + + - - ^ ^ ^ That tells you, by Descartes' Rule of Signs, that there are at most three positive real roots, and furthermore, that the number of positive real roots differs from three by an even number. That means that the number of positive real roots is either three or one (because 2 and 0 differ from 3 by an odd number). By considering: f(-x) = x^6 + 3*x^5 - 7*x^4 - 6*x^3 + 2*x^2 + x - 12 which also has three sign changes in the sequence of its coefficients, + + - - + + - ^ ^ ^ you see that f(-x) also has either three or one positive real roots, so f(x) has three or one negative real roots. That gives you four possible setups: Positive Negative Complex 3 3 0 3 1 2 1 3 2 1 1 4 (For the particular f(x) I gave above as an example, the last of these possibilities holds. For other examples, any of the other of these could happen.) - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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