Problem 61 - College Algebra

Question

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros of \(f(x)=\) \(x^{5}-x^{4}+x^{3}-x^{2}+x-8 .\) Verify your result by using a graphing utility to graph \(f\).

Step-by-Step Solution

Answer

Applying Descartes' Rule of Signs to the polynomial function \(f(x)=x^{5}-x^{4}+x^{3}-x^{2}+x-8\), we find that there are either 3 or 1 positive real zeros and either 1 or 0 negative real zeros. Confirm these results using a graphing utility.

Step 1: Applying Descartes' Rule of Signs for positive real zeros

Descartes' rule of signs states that the number of positive real roots of a polynomial function is equal to the number of changes in the sign of its coefficients, or less than that by a positive even integer.\nThe polynomial function given is \(f(x)=x^{5}-x^{4}+x^{3}-x^{2}+x-8\). One can see that the number of sign changes is 3: + to - between \(x^5\) and \(x^4\), - to + between \(x^4\) and \(x^3\), and + to - between \(x\) and -8. Therefore, the possible number of positive real zeros is 3 or less than that by a positive even integer (i.e., 3 or 1).

Step 2: Applying Descartes' Rule of Signs for negative real zeros

For finding the number of negative real zeros, we replace \(x\) with \(-x\) and apply the rule of signs again. The substituted polynomial is \(f(-x)=(-x)^{5}-(-x)^{4}+(-x)^{3}-(-x)^{2}+(-x)-8\), which simplifies to \(-x^{5}-x^{4}-x^{3}-x^{2}-x-8\). Counting the sign changes, it can be seen that there's only one sign change, between -x and -8. Therefore, the possible number of negative real zeros is 1 or less than that by a positive even integer (i.e., 1 or 0).

Step 3: Verifying the result using Graph

To confirm our results, use a graphing utility to plot the given polynomial function \(f(x)\). The points where the graph crosses the x-axis will give you your real zeros. Note down the number of positive and negative zeros from the graph and cross-check with the prediction made using Descartes' Rule of Signs. If both results match, then our analysis is correct. If they don't, check the calculations again.

Key Concepts

Understanding Polynomial Functions

A polynomial function is a mathematical expression consisting of variables, coefficients, and exponents arranged in terms of powers. It is generally represented in the standard form:

• F(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0

where:

• "n" is the highest degree of the polynomial.
• "a_n, a_{n-1}, ..., a_0" are constants known as coefficients.

Polynomial functions can have one or more terms. The specific exercise works with a fifth-degree polynomial:

\[ f(x) = x^5 - x^4 + x^3 - x^2 + x - 8 \]Seeing the variety of terms and powers in a polynomial allows us to explore concepts like solving for roots or zeros using special rules like Descartes’ Rule of Signs. Understanding this will help us analyze different aspects of any polynomial, including its behavior and graphical representation.

Exploring Real Zeros of a Polynomial

Real zeros of a polynomial function are the values of "x" that make the function equal to zero, meaning:

• F(x) = 0

Identifying these zeros essentially involves solving the equation for the given polynomial. Real zeros are important as they tell us where the graph of the polynomial intersects the x-axis. Descartes' Rule of Signs helps us predict the number of real zeros by examining the changes in the sign of the coefficients of the polynomial:

• Each sign change corresponds to a possible real zero.

For our polynomial:
Sign changes: Positive = 3 changes (possibly 3 or 1 positive real zeros), Negative = 1 change (possibly 1 or 0 negative real zeros).

By calculating these, we can create a framework for checking against actual graph results, creating a connection between theoretical math and visual interpretation.

Utilizing a Graphing Utility

A graphing utility is an essential tool for visualizing mathematical functions and their behaviors. These tools might be physical graphing calculators or software applications on a computer or online. To verify our theoretical findings using Descartes' Rule of Signs, we can graph our polynomial:

• \( f(x) = x^5 - x^4 + x^3 - x^2 + x - 8 \)

Here's how to use a graphing utility for this task:

• Enter the polynomial into the utility.
• Generate the graph and observe where the curve intersects the x-axis.
• Count the number of intersection points to find the real zeros.

This graphical approach provides a concrete visual representation, confirming our calculations and predictions about the zeros of the polynomial. It is a powerful method that reinforces the connection between algebraic techniques and their graphical manifestations.