Problem 36 - 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. $$f(x)=x^{4}+6 x^{3}-18 x^{2} ; \text { between } 2 \text { and } 3$$

Step-by-Step Solution

Answer

By using the Intermediate Value Theorem and noting that there is a change in sign between \(f(2)\) and \(f(3)\), we can conclude that that there is at least one real zero for the function \(f(x)=x^{4}+6 x^{3}-18 x^{2}\) between 2 and 3.

Step 1: Finding the value of function at 2

First find the value of the function at the lower limit by replacing \(x\) with 2 in the given function \(f(x)=x^{4}+6 x^{3}-18 x^{2}\):\n\n \(f(2)=2^{4}+6(2)^{3}-18(2)^{2}=16+48-72=-8\)

Step 2: Finding the value of function at 3

Now find the value of the function at the upper limit by replacing \(x\) with 3 in the given function \(f(x)=x^{4}+6 x^{3}-18 x^{2}\):\n\n \(f(3)=3^{4}+6(3)^{3}-18(3)^{2}=81+162-162=81\)

Step 3: Applying Intermediate Value Theorem

In order to apply the Intermediate Value Theorem, there should be a change in the sign of the function value at the given interval [a,b]. \n\n In this case, \(f(2)\) is negative and \(f(3)\) is positive, which means the function value changes sign from negative to positive. Therefore, by the Intermediate Value Theorem, there is at least one real zero between 2 and 3.

Key Concepts

Polynomials

Polynomials are mathematical expressions involving a sum of powers of variables with their coefficients. They take the form of expressions like \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \]where - \(a_n, a_{n-1}, \ldots, a_0\) are constants known as coefficients,- \(x\) is the variable.
Polynomials can have different degrees depending on the highest power of the variable. In the problem given above, the polynomial is \[ f(x) = x^4 + 6x^3 - 18x^2 \]which is of degree 4 because the highest exponent is 4. Higher degree polynomials can be more complex, having curves known as roots or zeros, where the polynomial evaluates to zero. Knowing the structure of a polynomial is crucial in analyzing its behavior, such as determining where its real zeros might be.

Real Zeros

Real zeros refer to the values of \(x\) that make a polynomial equal to zero. For any polynomial equation \(f(x) = 0\), the real solutions are the real zeros. These are the points where the graph of the polynomial crosses or touches the \(x\)-axis.
The Intermediate Value Theorem (IVT) is a handy tool to identify if a real zero exists within a specific interval. According to IVT, if a continuous function \(f(x)\) has opposite signs at two points \(a\) and \(b\) (i.e., \(f(a)\) and \(f(b)\)), then there must be at least one real zero between \(a\) and \(b\). In simpler terms, if \(f(x)\) changes from a negative to a positive within an interval, the function will cross the \(x\)-axis, indicating a real zero in that range.

Evaluating Functions

Evaluating functions involves replacing the variable in the polynomial with a specific number and calculating the result. This process helps determine the value of a polynomial at particular points, which is essential for applying the Intermediate Value Theorem.
For example, if we have the function \[ f(x) = x^4 + 6x^3 - 18x^2 \]and we wish to evaluate it at \(x = 2\), we substitute 2 for \(x\) in the expression:

• Calculate \(2^4 = 16\)
• Calculate \(6 \, \times \, 2^3 = 48\)
• Calculate \(-18 \, \times \, 2^2 = -72\)
• Add the results: \(16 + 48 - 72 = -8\)

Similarly, evaluating at \(x = 3\) would involve substituting 3 in place of \(x\) and carrying out the arithmetic to find \(f(3)\).
This straightforward yet important step allows us to use IVT by confirming different signs of \(f(x)\) at those interval endpoints.