Problem 36 - Algebra and Trigonometry

Question

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. \(f(x)=x^{4}+6 x^{3}-18 x^{2} ;\) between 2 and 3

Step-by-Step Solution

Answer

To show that the polynomial \(f(x)=x^{4}+6 x^{3}-18 x^{2}\) has a real zero between the values of 2 and 3, one can use the Intermediate Value Theorem. By evaluating the function at \(x = 2\) and \(x = 3\), one can determine if it crosses the x-axis between these two points, because the value of the function changes sign. If it does, then there is a zero in that interval.

Step 1: Identify the Function and the Interval

In this case, the function is given by \(f(x)=x^{4}+6 x^{3}-18 x^{2}\). The goal is to prove that there exists a root (or 'zero') of the function, which is a real number between 2 and 3.

Step 2: Evaluate the Function at the Endpoints

Evaluate the function at \(x = 2\) and \(x = 3\). So find \(f(2)\) and \(f(3)\). This will give the values of the function at these two points.

Step 3: Apply the Intermediate Value Theorem

After Step 2, if \(f(2)\) and \(f(3)\) have different signs, it means that the function has crossed the x-axis between the points where \(x = 2\) and \(x = 3\), because it has gone from having a positive to a negative value or vice versa. According to the Intermediate Value Theorem, this implies that there must be a zero between these two points. If \(f(2)\) and \(f(3)\) have the same sign, the theorem does not provide any insight, and there may or may not be a zero between these two points.

Key Concepts

Polynomial functions

Polynomial functions are an important class of mathematical functions, easily recognizable by their structure. They are composed of variables raised to whole-number exponents, multiplied by coefficients. A general polynomial function can be written as:

• \(f(x) = a_n x^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\)

In this expression, \(a_n, a_{n-1}, \ldots, a_1, a_0\) are coefficients, and \(n\) is the degree of the polynomial, determined by the highest exponent of \(x\).
This structure makes polynomials versatile and widely applicable in various branches of mathematics and science. In the given problem, the polynomial \(f(x) = x^4 + 6x^3 - 18x^2\) is of degree 4, indicating a potential for up to 4 real roots.
Understanding polynomial functions enables us to apply important mathematical theorems and principles, such as the Intermediate Value Theorem, to find zeroes and analyze their behavior.

Finding real zeros

Finding real zeros of a polynomial involves determining the values of \(x\) where the function equals zero. For our polynomial, this means finding the values of \(x\) for which \( x^4 + 6x^3 - 18x^2 = 0 \). This task can be simplified when using tools like the Intermediate Value Theorem.

• Identify the interval of interest, such as \([2, 3]\) in this problem.
• Evaluate the polynomial at the endpoints of the interval: calculate \(f(2)\) and \(f(3)\).
• Check if the evaluations yield opposite signs.

If \(f(2)\) and \(f(3)\) are of different signs, we deduce that a real zero exists between these endpoints. This variation in sign suggests a point where the function crosses the x-axis. Thus, applying the Intermediate Value Theorem can confirm the presence of a zero without solving the equation explicitly.

Function evaluation

Function evaluation is the process of determining the output of a function for specific input values. It is a fundamental concept in algebra and calculus, especially when analyzing the behavior of functions over certain intervals. In this exercise, evaluating the function \(f(x) = x^4 + 6x^3 - 18x^2\) at specific points like \(x = 2\) and \(x = 3\) helps us understand the function's behavior between these points.
Steps for function evaluation:

• Substitute the chosen value of \(x\) into the polynomial expression.
• Perform the arithmetic operations as per the polynomial's expression.
• Record the output, \(f(x)\), which reveals whether the function value is positive or negative.

For example, calculate \(f(2)\) and \(f(3)\) by substituting 2 and 3 into the polynomial. These calculated values are essential for applying the Intermediate Value Theorem, as they provide the necessary sign change indication to assess the existence of a real zero in the specified interval.