Problem 426 - Precalculus

Question

Use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph. $$f(x)=x^{3}-2 x^{2}-16 x+32$$

Step-by-Step Solution

Answer

2 or 0 positive roots, 1 or 0 negative roots; graph confirms 2 positive, 1 negative root.

Step 1: Identify Sign Changes for Positive Roots

The first step to apply Descartes' Rule of Signs is to identify how the signs change in the polynomial \( f(x) = x^3 - 2x^2 - 16x + 32. \) Look at the sequence of coefficients: \( [+1, -2, -16, +32] \). We can observe the sign changes from \(+1 \to -2\) and \(-16 \to +32\), resulting in 2 sign changes.

Step 2: Determine Possible Positive Roots

Based on the 2 sign changes, Descartes’ Rule of Signs tells us that there can be 2 or 0 positive real roots.

Step 3: Identify f(-x) for Negative Root Analysis

To determine the possible number of negative roots, evaluate \( f(-x) \). Substitute \(-x\) in place of \(x\) in the original polynomial: \[ f(-x) = (-x)^3 - 2(-x)^2 - 16(-x) + 32 = -x^3 - 2x^2 + 16x + 32. \] This results in the sequence: \([-1, -2, +16, +32] \).

Step 4: Identify Sign Changes for Negative Roots

In the transformed polynomial \( f(-x) = -x^3 - 2x^2 + 16x + 32 \), observe the sequence of coefficients \([-1, -2, +16, +32] \). There is one sign change from \(-2 \to +16\), indicating possibly 1 or 0 negative real roots.

Step 5: Graph Confirmation

Graph the polynomial \( f(x) = x^3 - 2x^2 - 16x + 32 \). Confirm by checking the graph that there are two positive roots and one negative root. You should see two intersections with the x-axis in the positive region and one in the negative region.

Key Concepts

Polynomial Roots

When we talk about the roots of a polynomial, we are referring to the values of \(x\) that make the polynomial equal zero. For the polynomial \(f(x) = x^3 - 2x^2 - 16x + 32\), its roots are the points where the graph of the equation crosses the x-axis. These are the solutions to \(f(x) = 0\).
Understanding polynomial roots is crucial because they give us insight into the behavior of polynomial functions. In this case, the roots tell us where the function reaches a value of zero and changes sign. This is especially useful when solving equations, as roots represent the solutions of the equation.

Positive Roots

Positive roots of a polynomial are the solutions where the x-values are greater than zero. To find the number of possible positive roots, we use Descartes' Rule of Signs on the original polynomial. For the polynomial \(f(x) = x^3 - 2x^2 - 16x + 32\), we observe the sequence of its coefficients:

• [+1, -2, -16, +32]

We note sign changes between \(+1 \to -2\) and \(-16 \to +32\). Each sign change indicates a possible positive root. Thus, there can be 2 or 0 positive roots for this polynomial.
Descarte's Rule of Signs helps us predict how many roots of a particular sign a polynomial might have. It’s important because it simplifies the complex problem of finding all potential roots without complex computations.

Negative Roots

Negative roots refer to the solutions of the polynomial where the x-values are less than zero. To determine them, we evaluate the polynomial at \(-x\):

• \( f(-x) = -x^3 - 2x^2 + 16x + 32 \)

By transforming the polynomial, we then examine the sequence of coefficients

• \([-1, -2, +16, +32]\)

Here, there is a single sign change from \(-2 \to +16\). According to Descartes' Rule of Signs, this results in 1 or 0 negative roots.
Understanding negative roots lets us see how the polynomial behaves for negative values and can be helpful for full graph analysis to predict polynomial behavior in different quadrants.

Graph Analysis

Graph analysis involves visually examining the graph of a polynomial to confirm the existence and number of roots. When you graph \(f(x) = x^3 - 2x^2 - 16x + 32\), the roots or solutions are visible as the x-intercepts.
The graph should intersect the x-axis at these points:

• Two intersections in the positive x-region indicate two positive roots.
• One intersection in the negative x-region shows one negative root.

Graphing the polynomial not only confirms the number of roots predicted by Descartes' Rule of Signs but also provides a clear, visual understanding of where those roots are located on the number line. It makes abstract algebraic concepts tangible and easier to understand.