Problem 23 - College Algebra

Question

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)=x^{3}+2 x^{2}+5 x+4 $$

Step-by-Step Solution

Answer

The possible number of positive real zeros of the function is 0 and the possible number of negative real zeros is 2.

Step 1: Identify the polynomial function

The given polynomial function is \(f(x) = x^{3} + 2x^{2} + 5x + 4\). It is important to note that the polynomial is ordered in descending powers of \(x\).

Step 2: Apply Descartes' Rule for positive real zeros

For positive real zeros, count the number of sign changes in the polynomial. Here, there are no sign changes as all the terms are positive. Therefore, the possible number of positive real zeros is 0.

Step 3: Apply Descartes' Rule for negative real zeros

For negative real zeros, replace \(x\) with \(-x\) and recount the changes. Thus, \(f(-x)=(-x)^{3}+2(-x)^{2} +5(-x)+4 = -x^{3} + 2x^{2} - 5x + 4\). Now, count the sign changes in this new polynomial. There are two sign changes (from \(-x^{3}\) to \(2x^{2}\) and from \(2x^{2}\) to \(-5x\)). Therefore, the possible number of negative real zeros is 2.

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers and multiplied by coefficients. These functions are often written in descending order of the power of the variable. For example, the function \( f(x) = x^{3} + 2x^{2} + 5x + 4 \) is a polynomial function of degree 3 because the highest power of \( x \) is 3. Polynomials are used to model a wide range of phenomena in science and engineering because they can describe curves that change in ways that align well with real-world data. When dealing with polynomial functions, some of the key aspects to consider include their degree, coefficients, and zeros.

Real Zeros

Real zeros of a polynomial are the values of \( x \) that make the polynomial equal to zero. These zeros correspond to the points where the graph of the polynomial crosses or touches the x-axis. Finding the real zeros is crucial in understanding the behavior of the polynomial, since each zero represents a point where the polynomial will change direction or level out.

• If a polynomial has a real zero at \( x = c \), then \( f(c) = 0 \).
• The number of real zeros is at most equal to the degree of the polynomial.

Knowing the real zeros of a polynomial helps in determining the roots of the equation and subsequently in solving algebraic equations.

Sign Changes in Polynomials

In the analysis of polynomial functions, observing sign changes helps identify the potential number of positive and negative real zeros. A sign change occurs when consecutive terms in a polynomial go from positive to negative or vice versa. Descartes's Rule of Signs specifically relies on these changes to make conclusions:

• Count the number of times the sign of the terms changes as you move from the highest power to the lowest.
• For the polynomial \( f(x) = x^{3} + 2x^{2} + 5x + 4 \), since all terms are positive, there are no sign changes, leading to an estimate of 0 positive real zeros.

Observing the sign changes after substitution helps in predicting the negative zeros as well, providing a fuller understanding of the polynomial's potential roots.

Positive and Negative Zeros Analysis

Descartes's Rule of Signs offers a method for analyzing the possible number of positive and negative real zeros in a polynomial. To determine the number of positive real zeros, count the sign changes in the polynomial as it is written. To find the likely number of negative zeros, substitute \( x \) with \( -x \) and count the sign changes again:

• For \( f(x) = x^{3} + 2x^{2} + 5x + 4 \), there are 0 sign changes, indicating 0 positive real zeros.
• Substituting \( -x \) yields the new polynomial \( f(-x) = -x^{3} + 2x^{2} - 5x + 4 \) with 2 sign changes, indicating at most 2 negative real zeros.

These insights help set the stage for further analysis using algebraic methods, or numerical approximations to find exact values for the polynomial's zeros.