Question

Use the Rational Zero Test to list all possible rational zeros of \(f\). Then use a graphing utility to graph the function. Use the graph to help determine which of the possible rational zeros are actual zeros of the function. $$f(x)=x^{3}+x^{2}-4 x-4$$

Short answer

The actual zeros of the polynomial function \(f(x)=x^{3}+x^{2}-4x-4\) are \(x=1\) and \(x=-2\).

Step-by-step solution

List All Possible Rational Zeros

Using the Rational Zero Test, we can list all possible rational zeros of the function \(f\). This can be done by taking the factors of the constant term (which is -4) and dividing them by the factors of the leading coefficient (which is 1). The constant term factors are \(\pm1, \pm2, \pm4\). The leading coefficient's factor is just 1. Hence, the possible rational zeros are \(\pm1, \pm2, \pm4\).

Sketching the Graph of the Function

Use a graphing utility to help visualize the function \(f(x)=x^{3}+x^{2}-4x-4\). The graph can also help in directly verifying whether the values from Step 1 are actually solutions of the equation \(f(x)=x^{3}+x^{2}-4x-4=0\). If it is a solution, then it should intersect the x-axis at that point.

Determine the Actual Zeros

Looking at the graph of \(f(x)\) found in Step 2, identify where the graph intersects the x-axis. These intersections represent the actual zeros of the polynomial function. According to the graph of the function, it can be observed that the function intersects the x-axis at \(x=1\) and \(x=-2\), therefore, the actual zeros of the function \(f\) are \(x=1\) and \(x=-2\).

Key concepts

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables and coefficients, combined using only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial function is given by the equation:

\( f(x) = x^{3} + x^{2} - 4x - 4 \).

One crucial aspect of understanding polynomial functions is knowing their degree, which is the highest exponent of the variable. The degree tells us the maximum number of possible zeros (places where the graph crosses or touches the x-axis) and turns (changes in the direction of the graph) the polynomial can have. For instance, the given function \( f(x) \) is a cubic polynomial, meaning it has a degree of three. This implies that the graph of the function can have at most three x-intercepts and two turns. Moreover, the leading coefficient, found in the term with the highest exponent, can affect the end behavior of the graph, which is how the line stretches or shrinks as it approaches infinity.

Graphing Utility

A graphing utility is a powerful tool for visualizing polynomial functions and identifying their key characteristics. Tools like graphing calculators or computer software can take a given polynomial equation and produce an accurate graph.

To illustrate with our example, \( f(x) = x^{3} + x^{2} - 4x - 4 \), plotting this equation using a graphing utility will help students to see where the function rises or falls, how it behaves as x approaches positive or negative infinity, and most importantly, where it crosses the x-axis.

The ability to graph a function is particularly crucial when it comes to identifying the actual zeros of a polynomial. By examining where the graph intersects the x-axis, students can confirm which potential zeros suggested by the Rational Zero Test are, in fact, real solutions. This visual approach complements the theoretical work and offers a concrete understanding of abstract concepts.

Actual Zeros

Actual zeros of a polynomial function are the values of x for which the value of the function is zero. These are the points where the graph of the polynomial touches or crosses the x-axis. Finding the actual zeros is a two-step process: first, we use the Rational Zero Test to hypothesize potential rational zeros, and then we use a graphing utility to confirm which of these are actual zeros.

In the discussed example, when we graph the function \( f(x) = x^{3} + x^{2} - 4x - 4 \), we look for intersections with the x-axis. The intersections found, such as \( x=1 \) and \( x=-2 \), are the actual zeros of this function. These points are essential not only for understanding the graph's shape but they also have numerous applications in real-world problems, such as physics, engineering, and economics, where finding the zeros of functions can represent equilibrium points, optimal values, or rates of change. Understanding how to identify and interpret these zeros is fundamental for students studying polynomial functions.