Problem 13 - College Algebra

Question

In Exercises \(7-14,\) show that each polynomial has a real zero between the given integers. Then use the Intermediate Value Theorem to find an approximation for this zero to the nearest tenth. If applicable, use a graphing utility's zero feature to verify your answer. \(f(x)=3 x^{3}-10 x+9 ;\) between \(-3\) and \(-2\)

Step-by-Step Solution

Answer

The polynomial \(f(x)= 3x^{3}- 10x+9\) has at least one real zero between \(-3\) and \(-2\). Using the Intermediate Value Theorem, this zero is approximated to be \(-2.5\).

Step 1: Evaluate the Polynomial at Each Interval Endpoint

First, evaluate the polynomial at each endpoint of the interval by substituting \(-3\) and \(-2\) into the function \(f(x)=3x^{3}- 10x+9\). Therefore, \(f(-3) = 3(-3)^{3}-10(-3)+9 = -6\), and \(f(-2) = 3(-2)^{3}-10(-2)+9 = 5\).

Step 2: Determine a Sign Change

Next, observe that \(f(-3)\) and \(f(-2)\) have different signs. This means that there is at least one real zero of the function between \(-3\) and \(-2\), because the function has to pass through zero to get from a negative to a positive value.

Step 3: Approximate the Zero

Now you need to approximate the zero of the function to the nearest tenth. Using the Intermediate Value Theorem, since the zero lies at some value within the interval from \(-3\) to \(-2\), and since the function changes signs at approximately equal intervals, it can be approximated that the zero lies halfway between \(-3\) and \(-2\). Thus, the zero might be at \(-2.5\).

Step 4: Verify with a Graphing Utility (Optional)

If a graphing utility is available, the zero feature can be used to verify the answer. Plotting the function and using the zero feature will show whether \(-2.5\) is indeed a root.

Key Concepts

Real Zero of a Polynomial

In mathematics, when we talk about the real zero of a polynomial, we're referring to the values of x for which the polynomial function evaluates to zero. Imagine a polynomial f(x) as a roller coaster, the ups and downs representing values of f(x) over the domain of x. A real zero is like those points where the roller coaster crosses the ground level, neither lifting off into the sky (positive values) nor tunneling underground (negative values).

From the exercise, by plugging in -3 and -2 into the polynomial f(x), we get two different signs. This is crucial because it indicates that there's at least one point in that interval where our polynomial 'roller coaster' must cross the ground level. In this context, the ground level represents zero on the y-axis, which means a real zero exists between -3 and -2.

Polynomial Function Evaluation

Evaluation of polynomial functions, like f(x) = 3x^3 - 10x + 9, involves plugging in specific values of x to find the corresponding f(x). This is akin to checking the height of our metaphorical roller coaster at different points along the track to determine its trajectory. In Step 1 of the solution, we evaluate the function at two points: -3 and -2. Doing this gives us the heights (or depths) of f(-3) = -6 and f(-2) = 5. It's like checking if the coaster is in the air or below ground, at two distinct points, to infer its path.

Evaluating these points helps to provide a before-and-after snapshot of where the function's graph crosses the x-axis. If one value is positive and the other is negative, we know that the function has crossed the x-axis somewhere between those two x-values. Therefore, this method is essential for predicting and confirming the existence of real zeros.

Graphing Utility Zero Feature

Modern technology assists in mathematics through tools like graphing utilities, which often have a zero feature. This feature allows users to visually identify the exact points where the function crosses the x-axis, which represents the real zeros of the function. Imagine that you've plotted your roller coaster on a computer program. By using the zero feature, the program can automatically point out those precise moments when the coaster is at ground level.

Referencing Step 4 of the solution, after we've made an educated guess that the zero is at -2.5 based on the Intermediate Value Theorem, we can use graphing software to check our work. By inputting our polynomial into this software and applying the zero feature, we skip the manual search and get a precise answer, verifying if our estimated zero is accurate or if we need to refine our approximation. This tool is especially helpful in complex functions where manual calculation and estimation may be difficult.