Problem 39 - Algebra and Trigonometry

Question

In Exercises \(33-40,\) use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$f(x)=3 x^{3}-10 x+9 ; \text { between }-3 \text { and }-2$$

Step-by-Step Solution

Answer

Upon calculation, \(f(-3) = -6\) and \(f(-2) = 15\). These two have different signs. Therefore, by using the Intermediate Value Theorem, we conclude that there is at least one real zero of the function \(f(x)\) between -3 and -2.

Step 1: Evaluating the function at -3

First, we should find the value of the function at -3. We substitute \(x = -3\) into the function: \(f(-3) = 3(-3)^3 - 10(-3) + 9\).

Step 2: Evaluating the function at -2

Next, find the value of the function at -2 by substituting \(x = -2\) into the function: \(f(-2) = 3(-2)^3 - 10(-2) + 9\).

Step 3: Applying the Intermediate Value Theorem

Based on the Intermediate Value Theorem, if \(f(-3)\) and \(f(-2)\) have different signs, then there is a real zero between -3 and -2.

Key Concepts

Real Zeros

Real zeros, also known as roots of a function, are the values of \(x\) for which the polynomial function equals zero. In other words, they are the solutions to the equation \(f(x) = 0\). Finding real zeros is essential because they represent the points where the polynomial graph intersects the x-axis.

• If a polynomial has a zero at \(x = a\), then the function value \(f(a) = 0\).
• Real zeros can help in graphing polynomial functions and understanding their behavior.
• They can be found using algebraic methods, graphically, or using aids like the Intermediate Value Theorem.

By checking for real zeros, we can see where and how a polynomial crosses the x-axis, showing the real number solutions.

Polynomial Function

A polynomial function is a type of mathematical function that involves only non-negative integer powers of \(x\). It is usually written in the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are coefficients.

• The highest power of \(x\) determines the degree of the polynomial, which indicates the number of roots and the shape of its graph.
• For instance, the function \(f(x) = 3x^3 - 10x + 9\) is a cubic polynomial due to the third degree \(x^3\).
• Characteristics of the polynomial, such as its degree and leading coefficient, influence the nature and number of its real zeros.

Understanding polynomial functions is crucial for solving equations and analyzing graphs.

Evaluating Functions

Evaluating a function means calculating the output for specific input values of \(x\). It involves substituting a given number into the polynomial and simplifying to find the result.

• In the example problem, we evaluate the function at \(x = -3\) and \(x = -2\) to determine \(f(-3)\) and \(f(-2)\).
• The purpose of evaluating is to find the function's value at certain points to assist in analyzing its behavior or confirming solutions.
• Knowing the value of a function at specific points can indicate the presence of zeros within an interval.

Evaluating is a key skill for interpreting mathematical functions, especially when using methods like the Intermediate Value Theorem.

Different Signs

The concept of different signs is integral when applying the Intermediate Value Theorem. This theorem states that if a continuous function (such as a polynomial) moves from a negative output to a positive output—or vice versa—between two points, a real zero must exist between these points.

• In simple terms, if \(f(a) < 0\) and \(f(b) > 0\), or \(f(a) > 0\) and \(f(b) < 0\), a zero exists between \(a\) and \(b\).
• For our function \(f(x) = 3x^3 - 10x + 9\), we check that \(f(-3)\) and \(f(-2)\) have different signs to conclude a zero exists between these values.
• This method is powerful because it confirms a zero without requiring the exact value immediately.

Understanding this concept helps in identifying intervals to locate real zeros, especially when the polynomial behavior changes between two points.