Importance of Constant Term in Descartes' Rule of Signs?
Date: 05/03/2006 at 17:52:10 From: Alice Subject: possible exception to Descartes rule of signs Any third degree polynomial must have at least one real root. Is it possible to disprove this using Descartes' rule of signs and the polynomial p(x) = x^3 + x? Using Descartes, p(x) has 0 positive roots and p(-x) has 0 negative roots. Therefore it must have three imaginary roots! But p(x) has integer coefficients, and the conjugate root theorem says all polynomials with integer coefficients must have an even number of imaginary roots. The original question was: Does a third degree polynomial exist that has no real zeros? My student answered p(x) = x^3 + x and proved it using Descartes' rule. All definitions of Descartes mention nothing about having to have a constant. You can disprove the example by factoring out the x, however I am curious as my student used this example to answer the question on a test.
Date: 05/03/2006 at 20:07:39 From: Doctor Peterson Subject: Re: possible exception to Descartes rule of signs Hi, Alice. I recently noticed that the text I am using fails to point out what is happening in a problem much like this. When the constant term is zero, at least one of the roots is ZERO, since you can factor out an x. Descartes' Rule of Signs mentions only POSITIVE and NEGATIVE roots! So it tells us that your equation has 0 positive roots and 0 negative roots, but that does not mean there are no REAL roots; 0 is a real number, and is a root. Perhaps we should add in "Peterson's Rule of Zeros": Count the number of missing terms starting at the constant to find the multiplicity of zero as a root, and now you have the full count of real roots. So your student was misusing Descartes by assuming that if there are no positive or negative roots, there are no real roots. In the example, p(x) = x(x^2+1) = x(x+i)(x-i) has one real root (zero) and two imaginary roots (i and -i). If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 05/03/2006 at 21:25:58 From: Alice Subject: Thank you (possible exception to Descartes rule of signs) Dr. Peterson - Thanks so much for pointing out what should've been obvious to me! I'm giving my student credit on the test anyway for making me go to the lengths of asking Dr. Math! Alice
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