Problem 427 - Precalculus

Question

Use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph. $$f(x)=2 x^{4}-5 x^{3}-5 x^{2}+5 x+3$$

Step-by-Step Solution

Answer

The function has 2 or 0 possible positive and negative roots.

Step 1: Identify Changes in Sign for Positive Roots

The first step is to find the number of sign changes in the function for positive roots. The given function is \(f(x) = 2x^4 - 5x^3 - 5x^2 + 5x + 3\). - Observing the coefficients: \(2, -5, -5, 5, 3\).- Determine the sign sequence: \(+, -, -, +, +\).Count the number of sign changes between consecutive terms: from \(+\) to \(-\), from \( - \) to \(+\), and from \( + \) to \( + \).Thus, there are 2 sign changes.

Step 2: Determine Possible Number of Positive Roots

According to Descartes' Rule of Signs, the number of positive real roots is equal to the number of sign changes or less than that number by an even number. In this case, with 2 sign changes, the possible number of positive real roots is 2 or 0.

Step 3: Identify Changes in Sign for Negative Roots

To find the number of sign changes for negative roots, evaluate \(f(-x)\). This replaces each instance of \(x\) in the polynomial with \(-x\): \[ f(-x) = 2(-x)^4 - 5(-x)^3 - 5(-x)^2 + 5(-x) + 3 = 2x^4 + 5x^3 - 5x^2 - 5x + 3 \]The coefficients for this expression are \(2, 5, -5, -5, 3\), leading to the sign sequence: \(+,+,-,-,+\).Count the number of sign changes: from \(+ \) to \(-\) and from \(-\) to \( + \).That makes 2 sign changes.

Step 4: Determine Possible Number of Negative Roots

Applying Descartes' Rule to \(f(-x)\), with 2 sign changes, the possible number of negative real roots is 2 or 0.

Step 5: Verify with Graph

Check the graph of the polynomial function \(f(x) = 2x^4 - 5x^3 - 5x^2 + 5x + 3\). Observing the intersections of the curve with the x-axis will confirm the actual number of positive and negative roots. If the graph intersects the x-axis twice on the positive side, this confirms 2 positive roots. If it intersects twice on the negative side, then there are 2 negative roots.

Key Concepts

Positive Roots

When we discuss positive roots, we're talking about the values of \( x \) that make the polynomial equal to zero, which are strictly greater than zero. In the context of Descartes' Rule of Signs, finding positive roots involves analyzing how the sign of the polynomial's coefficients change as you move from one term to the next.

For instance, with the polynomial function \( f(x) = 2x^4 - 5x^3 - 5x^2 + 5x + 3 \), we assess the coefficients to identify sign changes. Here, the sign sequence is \(+,-,-,+,+\), resulting in two sign changes.

Descartes' Rule indicates that the potential number of positive roots equals the number of sign changes or fewer by an even number. Therefore, this function could have 2 or 0 positive roots. Essentially, this rule helps us hypothesize the root structure, guiding further analysis or verification, such as graphing.

Negative Roots

Negative roots refer to the roots of the polynomial that are less than zero. To find these using Descartes' Rule of Signs, we need to substitute each \( x \) in the polynomial with \( -x \). This forms a new expression, \( f(-x) \).

For \( f(x) = 2x^4 - 5x^3 - 5x^2 + 5x + 3 \), substituting gives us \( f(-x) = 2x^4 + 5x^3 - 5x^2 - 5x + 3 \). The coefficients \( 2, 5, -5, -5, 3 \) yield a sign sequence of \(+,+,-,-,+\).

Through Descartes' Rule, this sequence shows 2 sign changes, indicating that there could be 2 or 0 negative roots. This step is crucial for understanding the potential negative solutions and confirming them requires further methods like graphing or algebraic tests.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Polynomial functions can have several attributes, one of which includes their roots or solutions, which are the values of \( x \) that make the expression zero.

For example, in the polynomial \( f(x) = 2x^4 - 5x^3 - 5x^2 + 5x + 3 \), it consists of variables \( x \) raised to different power levels. Each term represents a part of the polynomial, with the highest power indicating the degree of the polynomial.

Understanding polynomials is straightforward when you break it down:

• Degree: The highest power of the variable in the polynomial.
• Coefficients: The numbers in front of each term.
• Constant term: The number without a variable, here being \(+3\).

Polynomial equations are essential in finding roots or solutions, creating graphs, or analyzing mathematical models.

Sign Changes

Sign changes in a polynomial help us predict the number of possible positive or negative roots using Descartes' Rule of Signs. It's all about observing how the signs of coefficients change from one term to the next.

Consider the polynomial \( f(x) = 2x^4 - 5x^3 - 5x^2 + 5x + 3 \). When we analyze its coefficients \( 2, -5, -5, 5, 3 \), they provide a pattern of sign changes: from positive to negative, and negative to positive.

This function shows two sign changes overall, which indicates potential root possibilities. Calculating the sign changes after substituting with \(-x\) helps find the number of negative roots.

• Positive roots: Count sign changes in \( f(x) \).
• Negative roots: Substitute \( x \) with \( -x \) and count again.

Sign changes give us a first glance at what root structure might look like, simplifying the process of analyzing polynomial solutions by outlining potential answers that need verification.