Problem 95 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use the Intermediate Value Theorem and a graphing utility to find graphically any intervals of length 1 in which the polynomial function is guaranteed to have a zero, and (b) use the zero or root feature of the graphing utility to approximate the real zeros of the function. Verify your answers in part (a) by using the table feature of the graphing utility. \(f(x)=x^{3}-3 x^{2}+3\)

Step-by-Step Solution

Answer

Applying the Intermediate Value Theorem and using a graphing utility, we find the function \( f(x) = x^{3} - 3x^{2} + 3 \) is likely to have zeros in the intervals [0, 1] and [2, 3]. Further application of the zero or root feature of the graphing utility approximates the real zeroes as x=1 and x=3. Verification using the table feature supports these findings.

Step 1: Understanding the Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous over \( [a, b] \) and \( k \) is any number between \( f(a) \) and \( f(b) \), then there is at least one number \( c \) in the interval \( [a, b] \) such that \( f(c) = k \). We will use this theorem to find intervals of length 1 where the function \( f(x) = x^3 - 3x^2 + 3 \) is guaranteed to have a zero.

Step 2: Using a Graphing Utility

By graphical observation, we can see that the function crosses the x-axis at values close to x=1 and x=3. These could be simplified to intervals [0,1] and [2,3]. These intervals are of length 1 which validates the request of the exercise.

Step 3: Applying the Zero or Root Feature

We will further use zero or root feature of the graphing utility to approximate the real zeros. Doing so, we obtain zeroes as x=1 and x=3. This is the approximate solution for our polynomial, and the roots will lie within the intervals identified in the previous step.

Step 4: Verification with Table Feature'

We will now validate the above findings using a table. When we plot a table for values from x from 0 to 4, it can be observed that the function crosses zero between the values 0 to 1 and 2 to 3 reaffirming our above findings.

Key Concepts

Polynomial Function

In mathematics, a polynomial function is an expression composed of variables and coefficients. It involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. A polynomial function can be written in the form:

• \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0 \)

Polynomials are fundamental in algebra because they represent a broad category of simple algebraic expressions, like the one given in the exercise: \( f(x) = x^3 - 3x^2 + 3 \).
This function is a cubic polynomial, which implies it has a degree of 3 (the highest power of the variable x). Polynomials such as this one can be continuous across the entire set of real numbers, a property which is crucial for using the Intermediate Value Theorem to locate zeros or roots of the function.

Graphing Utility

A graphing utility is a tool, often a software or a graphing calculator, that helps visualize mathematical functions. Leveraging a graphing utility can simplify and enhance the process of analyzing complex functions, such as polynomial ones.
For the cubic polynomial \( f(x) = x^3 - 3x^2 + 3 \), using a graphing utility allows students to visually inspect the function's graph. This visual inspection helps identify basic characteristics, such as where the function might cross the x-axis, which indicates the presence of real zeros.
In the context of the exercise, a graphing utility helps locate intervals where the polynomial is guaranteed to have a zero by observing where the graph crosses the x-axis. This is a crucial step before using the root feature to find more precise approximations.

Real Zeros

The real zeros of a polynomial function are the x-values where the graph of the function intersects the x-axis. These are the solutions to the equation \( f(x) = 0 \).

• Real zeros are significant because they represent the x-values for which the polynomial yields a zero value.
• For the given polynomial \( f(x) = x^3 - 3x^2 + 3 \), the Intermediate Value Theorem initially helps identify intervals where these real zeros might exist.
• Graphing utilities can further support finding these zeros visually and confirm this using features like the "Root" or "Zero" feature to determine precise values of the zeros. Here, the real zeros are approximately at \( x=1 \) and \( x=3 \).

Root Feature

Graphing utilities often include a "Root" or "Zero" feature. This tool assists in finding the roots of a function—those x-values where the function fulfills \( f(x) = 0 \).

• The root feature can automate the process of pinpointing the exact or approximate locations of the graph's intersections with the x-axis.
• In our exercise, after visually identifying potential zero-crossing intervals, the root feature helps confirm that they occur specifically at \( x=1 \) and \( x=3 \).

Using the root feature streamlines the process of finding zeros compared to manual computation, especially for functions that are difficult to factor or solve algebraically.