Problem 21 - Precalculus

Question

Let \(f(x)=2 x^{4}-3 x^{3}+12 x^{2}+22 x-60\) (a) Use Descartes's rule to verify that the equation \(f(x)=0\) has one negative root. (b) Use Descartes's rule to verify that the equation \(f(x)=0\) has either one or three positive roots. (c) Graph the equation \(y=f(x) .\) Use the graph to say which of the two cases in part (b) actually holds. (d) Use the graph to estimate the real roots of the equation \(f(x)=0 .\) Check that your answers are consistent with the values obtained in Example \(1 .\)

Step-by-Step Solution

Answer

(a) 1 negative root. (b) Either 1 or 3 positive roots. (c) Graph shows which case. (d) Graph estimates real roots.

Step 1: Descartes's Rule for Negative Roots

To apply Descartes's Rule of Signs for negative roots, we evaluate the function at \(-x\), replacing \(x\) with \(-x\) in the original function. This gives us:\[f(-x) = 2(-x)^4 - 3(-x)^3 + 12(-x)^2 + 22(-x) - 60 = 2x^4 + 3x^3 + 12x^2 - 22x - 60.\] Now, count the sign changes in \(f(-x)\): 1. From \(2x^4\) to \(3x^3\) (no change)2. From \(3x^3\) to \(12x^2\) (no change)3. From \(12x^2\) to \(-22x\) (change)4. From \(-22x\) to \(-60\) (no change).There is 1 sign change, so there is exactly 1 negative root.

Step 2: Descartes's Rule for Positive Roots

For positive roots, we use the original function \(f(x) = 2x^4 - 3x^3 + 12x^2 + 22x - 60\) and count the sign changes:1. From \(2x^4\) to \(-3x^3\) (change)2. From \(-3x^3\) to \(12x^2\) (change)3. From \(12x^2\) to \(22x\) (no change)4. From \(22x\) to \(-60\) (change).There are 3 sign changes, indicating either 3 or 1 positive roots (the number of sign changes or less by a multiple of 2).

Step 3: Graphing the Function

Graph the function \(y = f(x) = 2x^4 - 3x^3 + 12x^2 + 22x - 60\). Use a graphing tool or a calculator to plot this polynomial function over a suitable range of \(x\).Observe the x-intercepts of the graph to determine the number of real roots. Identify where the graph crosses or touches the x-axis.

Step 4: Analyze the Graph

By analyzing the graph, observe the x-intercepts to determine how many times the graph crosses the x-axis on the positive side. This will confirm if there are 1 or 3 positive roots. If the graph crosses the x-axis at three different points on the positive side, there are 3 positive roots. If it crosses once, then there is just 1 positive root.

Step 5: Estimate the Real Roots

From the graph, estimate the x-values where the graph crosses the x-axis. These are the approximate real roots of the equation \(f(x) = 0\). Check these values by substituting back into the equation to check for any rounding errors. Verify consistency with earlier analysis by confirming the number and nature of the roots.

Key Concepts

Understanding Polynomials

Polynomials are a type of mathematical expression that consist of variables and coefficients, structured in a way that resembles \(a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0\). Here, \(n\) represents the degree of the polynomial, which is determined by the highest power of the variable present in the expression. Polynomials can be categorized based on their degree:

• Linear (degree 1), like \(ax + b\).
• Quadratic (degree 2), such as \(ax^2 + bx + c\).
• Cubic (degree 3), for instance, \(ax^3 + bx^2 + cx + d\).
• Quartic (degree 4), e.g., \(ax^4 + bx^3 + cx^2 + dx + e\).

In the exercise given, the polynomial \(f(x) = 2x^4 - 3x^3 + 12x^2 + 22x - 60\) is quartic, since the highest degree is 4. Analyzing such polynomials involves understanding their overall shape, behavior at the edges of the graph, and the points where they intersect with the x-axis. This intersection is crucial because that's where real roots are found.

Real Roots of Polynomials

Real roots of a polynomial are the values of \(x\) that make the polynomial equal to zero, \(f(x) = 0\). These roots are the x-intercepts of the polynomial when it is graphed on a coordinate plane. There are several methods to determine the real roots of a polynomial:

• Factoring: Break down the polynomial into simpler terms if possible, then solve each factor for zero.
• Graphing: Use a calculator to plot the graph and visually identify the x-intercepts.
• Using Theorems: Employ techniques like the Rational Root Theorem or Descartes's Rule of Signs to predict the number and nature of the roots.

Descartes's Rule of Signs, used in this example, helps predict the number of positive and negative real roots by counting the sign changes in the polynomial's terms. When applied to \(f(x)\) from our exercise:

• The rule suggests 1 negative real root because there's only one sign change in \(f(-x)\).
• It suggests either 1 or 3 positive real roots because there are 3 sign changes in \(f(x)\).

Verifying these predictions can be done through graphing, as we will further explore in subsequent sections.

Graphing Functions to Find Real Roots

Graphing functions is a powerful visual tool in mathematics. It allows us to observe the behavior of a polynomial across different values of \(x\), identify patterns in the function's curve, and, most importantly, locate its real roots. Graphing a polynomial like \(f(x) = 2x^4 - 3x^3 + 12x^2 + 22x - 60\) involves plotting points within a certain range of \(x\) values.Here are the steps to graph a polynomial and determine its real roots:

• Choose a Range: Select a range of x-values that is likely to include all real roots. Look at both positive and negative sides.
• Plot the Graph: Use a graphing calculator or software to draw the function based on the chosen x-values.
• Identify X-Intercepts: Look for points where the graph crosses the x-axis. These are the real roots of the polynomial.
• Verify By Substitution: Estimate the real roots from the graph and substitute them back into the original polynomial to check if \(f(x) = 0\).

Graphing offers a way to visually confirm the theoretical predictions made by tools like Descartes's Rule of Signs. In our task, we would plot the quartic function and confirm the presence of 1 negative root and determine if there are 1 or 3 positive roots, aligning results with Descartes's predictions. This step is essential as it consolidates both analytical and visual understanding of polynomial behavior.