Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. \(f(x)=-2 x^{3}-x,\) between \(x=-1\) and \(x=1\).

Step-by-Step Solution

Answer

By IVT, there is at least one zero in the interval \((-1, 1)\).

Step 1: Understanding the Intermediate Value Theorem

The Intermediate Value Theorem (IVT) states that if a function \( f \) is continuous on a closed interval \( [a, b] \), then for any value \( N \) between \( f(a) \) and \( f(b) \), there exists at least one \( c \) in \( (a, b) \) such that \( f(c) = N \). In particular, if \( f(a) \) and \( f(b) \) have opposite signs, there is at least one zero in the interval \( (a, b) \).

Step 2: Check Continuity of the Function

The given polynomial \( f(x) = -2x^3 - x \) is a polynomial, and all polynomials are continuous over all real numbers. Therefore, \( f(x) \) is continuous on \([-1, 1]\).

Step 3: Evaluate the Function at the Endpoints

Calculate \( f(-1) \) and \( f(1) \). For \( x = -1 \), \( f(-1) = -2(-1)^3 - (-1) = 2 + 1 = 3 \). For \( x = 1 \), \( f(1) = -2(1)^3 - 1 = -2 - 1 = -3 \).

Step 4: Determine Sign Change on the Interval

At \( x = -1 \), \( f(-1) = 3 \), and at \( x = 1 \), \( f(1) = -3 \). Since \( f(-1) \) and \( f(1) \) have opposite signs, there is a sign change over \([-1, 1]\).

Step 5: Apply the Intermediate Value Theorem

Since \( f(x) \) is continuous on \([-1, 1]\) and there is a sign change between \( f(-1) \) and \( f(1) \), the Intermediate Value Theorem guarantees that there is at least one zero within the interval \((-1, 1)\).

Key Concepts

Polynomial Functions

Polynomial functions are a fundamental concept in mathematics. They consist of variables and coefficients, constructed using only addition, subtraction, and multiplication. Each term in a polynomial is composed of a coefficient and a variable raised to a non-negative integer exponent.

The general form of a polynomial function is given by:

• A polynomial function of degree \( n \) is expressed as \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_1, a_0 \) are constants, and \( a_n eq 0 \).

Polynomials can have varying degrees, depending on the highest exponent of the variable. For example, \(-2x^3 - x\) is a polynomial function of degree 3, with coefficients \(-2\) and \(-1\) for terms \(x^3\) and \(x\), respectively.

Polynomials are crucial as they are the building blocks of algebra and calculus. They have wide applications in modeling real-world scenarios, thanks to their straightforward and predictable behavior. Studying polynomial functions helps students understand more complicated functions and mathematical concepts.

Continuous Functions

A function is said to be continuous if it is smooth and unbroken throughout its domain. Continuity is an essential property because it allows for the application of various mathematical theorems and techniques, such as the Intermediate Value Theorem.

For a function to be continuous, there must be no gaps, jumps, or points of discontinuity. This means:

• The function is defined for every point within the interval.
• The limit of the function as it approaches any given point equals the function's value at that point.

Polynomial functions, like the one given in the exercise \(-2x^3 - x\), are naturally continuous over all real numbers. This is because they are constructed using basic arithmetic operations (addition, subtraction, and multiplication), which do not introduce any discontinuities.

Understanding the concept of continuous functions is key to grasping more advanced topics in calculus. It forms the foundation for analyzing limits, derivatives, and integrals, all of which rely heavily on the idea of smooth and unbroken behavior of functions over given intervals.

Zeros of Functions

The zeros of a function, also known as roots or solutions, are the values of the variable that make the function equal zero. They are crucial for understanding the behavior of functions, especially when solving equations or analyzing graphs.

To find the zeros of a polynomial function, one typically sets the function equal to zero and solves for the variable:

• For the polynomial \( f(x) = -2x^3 - x \), finding the zeros involves solving the equation \(-2x^3 - x = 0\).

These points where the graph of the function intersects the horizontal axis are of particular interest. The Intermediate Value Theorem, as shown in the exercise, helps identify the existence of zeros within specific intervals by confirming a sign change between the endpoints.

Understanding zeros or roots is vital in topics such as factoring polynomials, solving quadratic equations, and addressing real-world problems modeled by polynomial equations. By recognizing these concepts, students can tackle various algebraic problems and deepen their comprehension of mathematical relationships.