Proofs of Descartes Rule & Polynomial Roots in College Algebra Texts

In summary, Descartes Rule, also known as the Rule of Signs, is a method used to determine the number of positive and negative roots of a polynomial equation. This rule works by counting the sign changes in the coefficients of the polynomial. It is important in college algebra as it provides a systematic way to find roots and make solving polynomial equations more efficient. However, it has limitations such as only being applicable to polynomials with real coefficients and not being able to determine the exact values of the roots.
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A few of the (assumed to be good) textbooks on College Algebra discuss the use of Descartes Rule of Signs, and a test for upper and lower bounds for real zeros for polynomial functions; but these ideas are never proved in the books that I found. In which college course, and in which Mathematics textbooks are proofs for these ideas found?
 
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Only rigorous proof i found, it goes for 7 pages.

http://homepage.smc.edu/kennedy_john/POLYTHEOREMS.PDF [Broken]
 
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Proofs for Descartes Rule of Signs and the test for upper and lower bounds for real zeros for polynomial functions can typically be found in advanced courses in algebra, such as Abstract Algebra or Advanced Algebra. These proofs are not typically included in introductory college algebra textbooks because they require a deeper understanding of algebraic concepts and techniques.

In terms of specific textbooks, proofs for these ideas can be found in textbooks such as "Contemporary Abstract Algebra" by Joseph Gallian or "A First Course in Abstract Algebra" by John Fraleigh. These textbooks are commonly used in advanced algebra courses at the college level.

It is important to note that while these proofs may not be included in introductory college algebra textbooks, they are still important concepts that are used and applied in higher level math courses. Therefore, it is important for students to strive for a deeper understanding of these ideas and seek out resources, such as advanced algebra textbooks, to fully grasp their significance and applications.
 

What is Descartes Rule in college algebra?

Descartes Rule, also known as the Rule of Signs, is a method used to determine the number of positive and negative roots of a polynomial equation. It states that the number of positive roots is equal to the number of sign changes in the coefficients, and the number of negative roots is equal to the number of sign changes in the coefficients of the polynomial with its terms written in descending order.

What is a proof of Descartes Rule?

A proof of Descartes Rule involves using mathematical induction to show that the rule holds true for all polynomial equations. This involves showing that the rule works for a specific polynomial, and then showing that if it works for a polynomial of degree n, it also works for a polynomial of degree n+1.

How is Descartes Rule used to find the roots of a polynomial equation?

To find the roots of a polynomial equation using Descartes Rule, you need to first determine the number of positive and negative roots by counting the sign changes in the coefficients. Then, you can use the Rational Root Theorem or synthetic division to test possible roots until you find the correct ones.

Why is Descartes Rule important in college algebra?

Descartes Rule is important in college algebra because it provides a systematic way to determine the number of possible roots of a polynomial equation. This can help to narrow down the search for the roots and make solving polynomial equations more efficient.

Are there any limitations to Descartes Rule?

Yes, there are limitations to Descartes Rule. It only applies to polynomial equations with real coefficients, and it cannot determine the actual values of the roots. It can also only be used for polynomials with terms written in descending order, so some rearranging may be necessary.

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