Problem 423 - Precalculus

Question

Use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph. $$f(x)=x^{3}-2 x^{2}+x-1$$

Step-by-Step Solution

Answer

There are 1 or 3 possible positive roots, 0 negative roots.

Step 1: Identify the polynomial

We begin by examining the given polynomial: \( f(x) = x^3 - 2x^2 + x - 1 \). Our task is to use Descartes' Rule of Signs to determine the possible number of positive and negative real roots.

Step 2: Count sign changes for positive roots

To find the number of positive roots, we consider the original polynomial \( f(x) \). The coefficients are 1, -2, 1, and -1. Observing the signs, we have: \(+, -, +, -\). The signs alternate three times: from \(1 \) to \(-2\), from \(-2 \) to \(1\), and from \(1 \) to \(-1\). Thus, there are 3 sign changes. This tells us there could be 3 or 1 positive roots.

Step 3: Construct polynomial for negative roots

To determine the number of negative roots, we consider \( f(-x) \). Replacing \( x \) with \(-x\), we get \( f(-x) = (-x)^3 - 2(-x)^2 + (-x) - 1 = -x^3 - 2x^2 - x - 1 \). The coefficients are -1, -2, -1, and -1.

Step 4: Count sign changes for negative roots

Looking at the polynomial \( f(-x) = -x^3 - 2x^2 - x - 1 \), the coefficients are all negative \(-, -, -, -\). There are no sign changes. This indicates that there are 0 negative roots.

Step 5: Confirm with graph

To confirm, we graph \( f(x) = x^3 - 2x^2 + x - 1 \). The graph shows the locations of the real roots. By checking visually, we expect to find 1 real positive root, in line with the possibility of having 1 positive root as indicated in Step 2.

Key Concepts

Polynomial Roots

A polynomial is a mathematical expression consisting of variables and coefficients, combined only by addition, subtraction, multiplication, and non-negative integer exponents. The roots of a polynomial are the values of the variable that make the polynomial equal to zero. Essentially, these roots are the solutions to the polynomial equation.
Understanding polynomial roots is important because they reveal where the graph of the polynomial intersects the x-axis. For example, in the polynomial \( f(x) = x^3 - 2x^2 + x - 1 \), the roots tell us the x-values where the graph crosses or touches the x-axis.

• Real Roots: These are the roots that can be plotted on a real number line. They result from the polynomial equating to zero at specific x-values.
• Complex Roots: These roots involve imaginary numbers and do not appear as x-intercepts on a real coordinate graph.

Using Descartes' Rule of Signs helps us predict the number of positive and negative real roots in a polynomial. Knowing the roots is useful for solving equations, factorizations, or understanding the behavior of the polynomial's graph.

Positive Roots

Positive roots of a polynomial are those that are greater than zero. Finding out how many positive roots a polynomial has can sometimes be complex, but Descartes' Rule of Signs provides a helpful method.
For finding the positive roots using this rule:

• Identify the original polynomial, such as \( f(x) = x^3 - 2x^2 + x - 1 \).
• Observe the signs of the coefficients in order from highest to lowest power: \, \( +, -, +, - \).
• Count the number of sign changes. Here, the sign changes three times, indicating up to 3 or 1 positive roots.

The rule tells us that the number of positive roots could be equal to the number of sign changes, or less by an even number. For our polynomial, we see 3 sign changes, suggesting there could be either 3 positive roots or 1 positive root due to possible multiple occurrences of a single positive root.

Negative Roots

Negative roots are those that are less than zero. To examine the potential for negative roots in a polynomial, Descartes' Rule of Signs is again employed but with a slight variation.

• First, substitute \(-x\) for \(x\) in the polynomial to get \(f(-x)\).
• For the polynomial \( f(x) = x^3 - 2x^2 + x - 1 \), substituting \(-x\) gives \( f(-x) = -x^3 - 2x^2 - x - 1 \).
• By examining the signs of this new polynomial, \,\(-, -, -, -\), you notice that there are no sign changes.

This lack of sign changes indicates that there are zero negative roots according to Descartes' Rule. This suggests that the graph of the polynomial will not cross the x-axis at any negative x-value, providing essential information about the polynomial's behavior across different ranges of x. It confirms that the roots are located elsewhere, and likely members of the positive or complex category.