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Chapter 3: Problem 67

Describe how to use Descartes's Rule of Signs to determine the possible number of positive real zeros of a polynomial function.

Short Answer

Expert verified
Using Descartes's Rule of Signs, the number of positive real zeros in a polynomial function can be found by counting the number of sign changes between consecutive coefficients. The number of positive real zeros is either equal to the number of these sign changes, or less than this by an even number.

Step by step solution

01

Understand Descartes's Rule of Signs

Descartes's Rule of Signs is a tool used to predict the number of positive real zeros in a polynomial. The rule states that the number of positive real zeros is either equal to the number of sign changes between consecutive coefficients, or it is less than this by a constant even digit. A zero change is defined as a positive coefficient followed by a negative one or vice versa.
02

Identify the coefficients and their signs

The coefficients are the numerical parts of the terms in the polynomial. If the polynomial function is arranged in descending power, i.e., \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), the number before each variable \( x \) and the constant term are the coefficients. Each coefficient has a sign which can be positive (+) or negative (-). Identify these signs for each term in the polynomial.
03

Count the sign changes

Once the signs of coefficients have been identified, now search for the sign changes. A sign change happens when a positive coefficient is followed by a negative or vice versa. If the signs of two consecutive coefficients are the same, it means there is no sign change.
04

Apply Descartes's Rule of Signs

Use the number of sign changes to predict the number of positive real zeros. The number of positive real zeros is either equal to the number of sign changes or less than this by an even number. Thus, based on this rule, the possible number of positive real zeros can be predicted.

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