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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 

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!

Associated Topics:
High School Polynomials

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