Problem 9 - College Algebra

Question

In Exercises \(7-14,\) show that each polynomial has a real zero between the given integers. Then use the Intermediate Value Theorem to find an approximation for this zero to the nearest tenth. If applicable, use a graphing utility's zero feature to verify your answer. \(f(x)=2 x^{4}-4 x^{2}+1 ;\) between \(-1\) and 0

Step-by-Step Solution

Answer

The polynomial \(f(x)=2 x^{4}-4 x^{2}+1\) has a real zero between -1 and 0. The approximation of the root to the nearest tenth using Intermediate Value Theorem should be calculated as the procedure outlined in Step 3

Step 1: Calculate the function values at the endpoint

Calculate the values of the function \(f(x)=2 x^{4}-4 x^{2}+1\) at the endpoints -1 and 0. So, calculate \(f(-1)\) and \(f(0)\)

Step 2: Prove existence of a root in the range using Intermediate Value Theorem

If \(f(-1)\) and \(f(0)\) have different signs, there exists a root between these two values according to the Intermediate Value Theorem.

Step 3: Approximate the root

Find the mid-value of the range (-1,0) and calculate function's value at it. Continue to reduce the range using Intermediate Value Theorem until the decimal approximation of the root is found to the nearest tenth.

Key Concepts

Real Zeros of Polynomials

Polynomials are mathematical expressions involving a sum of powers of variables multiplied by coefficients. Finding the real zeros of a polynomial, also known as roots, means determining the values of the variable that make the polynomial equal to zero. These roots are crucial because they represent the points where the polynomial crosses the x-axis on a graph.
To find real zeros, consider the equation form of a polynomial:

• A polynomial, like our example function \( f(x) = 2x^4 - 4x^2 + 1 \), contains different terms with varying powers of \( x \).
• Real zeros are the solutions to \( f(x) = 0 \).
• Analyzing the function helps identify intervals where real zeros might exist.

The Intermediate Value Theorem aids in this process by providing a mathematical way to verify if a root is present within a certain interval. If a polynomial changes signs over an interval, then there's at least one real zero in that interval. This tool is indispensable for determining whether real zeros exist between given integers.

Root Approximation

After confirming the existence of a real zero, the next step is to approximate its value. The Intermediate Value Theorem is once again used here, to narrow down intervals where the sign change occurs until it gets as close as possible to the actual root. This process involves:

• Calculating the function's values at the midpoint of an interval to check for sign changes.
• Reducing the interval step-by-step by selecting new midpoints and continuing until the approximate value reaches the desired precision, such as to the nearest tenth.

The more rigorous the process, the closer the approximation gets to the true root. This process is called bisection and is effective for finding roots in small intervals. By using this method meticulously, one can arrive at a suitable approximation of the polynomial's root.

Graphing Utilities

Graphing utilities are technological tools that help visualize and compute various characteristics of functions, including roots. They provide a visual representation of where the polynomial intersects with the x-axis, confirming the presence and approximate location of real zeros. This makes learning about zeros and their approximations much easier.

• Within graphing utilities, a dedicated "zero" or "root" feature simplifies finding x-intercepts, often with greater speed and accuracy than manual calculations.
• By inputting a polynomial function, these tools efficiently display its graph, pinpointing zeros and offering numerical approximations.
• Checking manual approximations with graphing utilities ensures accuracy, reinforcing confidence in the solutions.

These capabilities underscore the importance of graphing utilities in both educational settings and more complex problem solving. They act as a bridge between theoretical calculations and visual understanding, aiding students significantly as they work through polynomial root problems.