Problem 79 - College Algebra

Question

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(79-82 .\) Then determine the number of real zeros and the number of imaginary zeros for each function. $$ f(x)-x^{3}-6 x-9 $$

Step-by-Step Solution

Answer

To find the exact number of real and imaginary zeros, you need to plot the function first and count the intercepts with x-axis. Subtract this count from the degree of the function (3 in this case) to obtain the number of imaginary zeros.

Step 1: Plotting the Function

Use a graphing utility to plot the given function \(f(x) = x^{3}-6x-9\). The points where the graph intersects the x-axis are the real zeros of the function.

Step 2: Identifying the Real Zeros

Interpreting the graph, identify the points where the plot crosses the x-axis. These are the real roots of the function.

Step 3: Calculating the number of Imaginary Zeros

The number of real and imaginary zeros of a polynomial function is equal to its degree. Since the given function is of degree 3, and if we denote the number of real zeros as 'r' from step 2, we can calculate the number of imaginary zeros as 3 - r.

Key Concepts

Real Zeros of a Polynomial Function

When exploring polynomial functions, understanding real zeros is crucial. Real zeros, also known as real roots, are the x-values where a polynomial equation equals zero and the graph of the function intersects the x-axis. For the polynomial equation \(f(x) = x^3 - 6x - 9\), these values are found graphically using a graphing utility or algebraically through factoring, if possible, or using methods like the Rational Root Theorem.

To visually determine the real zeros, plot the polynomial using a graphing tool which displays intersections with the x-axis. If you are working without technology, you would attempt to solve for x when \(f(x)=0\). In this exercise, after plotting using the graphing utility, the points of intersection reveal the exact number of real zeros. These x-values are where the function’s output is zero, hence ‘zeros’ of the function.

Imaginary Zeros of a Polynomial Function

Polynomials may also have imaginary zeros, which are not as easily visualized as real zeros. Imaginary zeros occur in complex conjugate pairs which means if \(a+bi\) is a zero, then \(a-bi\) is also a zero where \(i\) is the imaginary unit. The presence of imaginary zeros is indicated when the graph does not intersect the x-axis enough times to account for the polynomial's degree.

For example, if a cubic polynomial function like \(f(x) = x^3 - 6x - 9\) has only one real zero identified through the graphing utility, it must have two imaginary zeros, as the Fundamental Theorem of Algebra assures us that the total number of zeros (counting multiplicity and including both real and imaginary) will be equal to the degree of the polynomial. If the polynomial's graph crosses the x-axis fewer times than its degree, the difference indicates the number of imaginary zeros.

Using a Graphing Utility

A graphing utility is an instrumental resource in understanding polynomial functions. It serves as a visual aid that can graphically represent complex equations, making it easier to identify characteristics such as intercepts, turning points, and particularly the zeros of the function. To effectively use it for functions like \(f(x) = x^3 - 6x - 9\), enter the polynomial expression into the utility and observe the rendered graph.

Look for the points where the graph crosses the x-axis; these are indicators of the real zeros. Keep in mind that graphing utilities might not show all details if the viewing window is not appropriately set, so make sure to adjust it for a complete view of the graph. Also, be aware that graphing utilities will only show real zeros visually, and imaginary zeros must be deduced through analysis of the polynomial's degree relative to the identified real zeros.