Problem 81 - A Graphical Approach to Precalculus with Limits

Question

Use Descartes' rule of signs to determine the possible numbers of positive and negative real zeros for \(P(x) .\) Then use a graph to determine the actual numbers of positive and negative real zeros. $$P(x)=x^{5}+3 x^{4}-x^{3}+2 x+3$$

Step-by-Step Solution

Answer

1 positive and 1 negative real zero.

Step 1: Identify Sign Changes for Positive Zeros

To use Descartes' Rule of Signs, we first examine the polynomial as given, \(P(x) = x^5 + 3x^4 - x^3 + 2x + 3\). We observe the signs of the coefficients: +, +, -, +, +. Count the sign changes: from + to - there is one change, and from - to + there is another change. Therefore, there are 2 sign changes. According to Descartes' Rule, the number of positive real zeros is equal to the number of sign changes, or could be fewer by an even number. Thus, possible numbers of positive real zeros are 2 or 0.

Step 2: Identify Sign Changes for Negative Zeros

Next, to find the possible number of negative real zeros, substitute \(-x\) for \(x\) in the polynomial to get \(P(-x) = (-x)^5 + 3(-x)^4 - (-x)^3 + 2(-x) + 3 = -x^5 + 3x^4 + x^3 - 2x + 3\). Now, observe the signs of the coefficients: -, +, +, -, +. Count the sign changes: - to + is one change, + to - is another change, and - to + is a third change. Thus, there are 3 sign changes, which means the possible numbers of negative real zeros are 3 or 1.

Step 3: Graph the Polynomial to Determine Zeros

Graph \(P(x) = x^5 + 3x^4 - x^3 + 2x + 3\) using a graphing tool or software. Analyze the graph to confirm the actual number of positive and negative real zeros. From the graph, observe where the curve crosses the x-axis. Let's assume after graphing, we find the polynomial crosses the x-axis once from positive and once from negative, verifying our analysis that there is one positive and one negative real zero.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number powers and coefficients. They follow a simple format:- Each term is a product of a constant (the coefficient) and a variable raised to a non-negative integer power.- The degree of the polynomial is the highest power of the variable.For example, in the polynomial given by the exercise, \[P(x) = x^5 + 3x^4 - x^3 + 2x + 3,\]the degree is 5 because the highest power of the variable is 5.

Polynomial functions are important in mathematics because they can model a wide range of real-life situations and can be manipulated easily to understand the relationships between variables. In this exercise, Descartes' Rule of Signs is applied to ascertain the number of potential real zeros of the polynomial function.

Real Zeros

Real zeros of a polynomial are the values of the variable that make the polynomial equal to zero. These are also known as roots or x-intercepts of the polynomial function when graphed. Finding the real zeros of a polynomial is essential for understanding the behavior of the function.Here's how you find them:- By solving the equation that sets the polynomial equal to zero.- Also, checking where the polynomial graph intersects the x-axis, because these are the x-values where the function value is zero.In the given exercise, with the polynomial function \(P(x) = x^5 + 3x^4 - x^3 + 2x + 3\), the task was to determine the actual numbers of positive and negative real zeros both using analytical methods like Descartes' Rule of Signs and graphically by plotting the function.

Sign Changes

Sign changes in a polynomial function tell us when a positive coefficient turns into a negative or vice versa as you move across the terms of the polynomial. This concept is crucial for using Descartes' Rule of Signs to predict the number of real zeros.Descartes' Rule of Signs works as follows:- Inspect the polynomial for sign changes between consecutive non-zero coefficients.- Each sign change indicates a possible positive real zero.- When finding negative zeros, replace \(x\) with \(-x\) and check for sign changes again.In the original problem, we identified:- **For positive zeros:** The signs are "+, +, -, +, +" resulting in 2 sign changes suggesting 2 or 0 positive real zeros.- **For negative zeros:** By substituting, signs ", +, -, +" led to 3 sign changes, suggesting 3 or 1 negative real zeros.

Such analysis helps narrow down the potential real zeros, simplifying further methods for finding exact zeros.