Question

Using complete sentences, explain how the outcome of the Rational Root Theorem and Descartes’ Rule of Signs differ from one another

Answer 1

One of these theorems tells you what the roots are, if there are any roots of that type. The other theorem tells you how many positive and negative roots you can expect.

The Rational Root Theorem tells you how to make a list of root candidates that may be roots of the polynomial. Yet, there is no guarantee than any of the candidates will be rational roots. You have to test the candidates.

Descartes' Rule of Signs tells you in a general sort of way how many Real roots, positive and negative, to expect but does not name them.

Using the RR Theorem and Descartes' Rule of Signs in conjunction sometimes teases out more information about the roots of a given polynomial than either theorem alone would.