Problem 58 - College Algebra

Question

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(58-61 .\) Then determine the number of real zeros and the number of nonreal complex zeros for each function. $$ f(x)=x^{3}-6 x-9 $$

Step-by-Step Solution

Answer

The number of real zeros and the number of nonreal complex zeros can be determined after graphing the function and counting the point of intersections with x-axis and subtracting it from total number of zeros possible.

Step 1: Graph the polynomial function

First plot the polynomial function using a graphing tool. The function \(f(x) = x^{3} - 6x - 9\) can be plotted easily on many mathematical software or online tools.

Step 2: Determine the real zeros

From the graph, it can be observed that the real zeros are the points where the graph intersects the x-axis. Determine those points by carefully observing the graph.

Step 3: Determine the nonreal complex zeros

A complex zero of a polynomial is a complex number that makes the polynomial equal to zero. For a polynomial of degree n, if you already know the real zeros(denoted as k) and every complex zero has its conjugate also as a zero, then the number of nonreal complex zeros is equal to n - k. Polynomial of degree 3 always has 3 zeros, they can be real or complex.

Key Concepts

Graphing Polynomial Functions

The graphing of polynomial functions is a foundational skill in algebra that aids in understanding the nature of these functions. When graphing a polynomial, we're looking for several key features, such as the location of its zeros, the end behavior of the graph, and any turning points indicative of local maxima or minima. Specifically, tools like graphing calculators enable students to plot complex functions and ascertain these characteristics visually.

Begin by establishing a coordinate system and plotting any known points, such as the y-intercept. In the case of the given polynomial function, we would observe the curve's behavior as it increases or decreases, and how it responds as x approaches positive or negative infinity. With a polynomial of degree three, such as this example function, the graph will generally display up to two turning points and exhibit end behaviors opposite in direction.

Real Zeros

Real zeros, also known as x-intercepts, are the points where the graph of a polynomial crosses the x-axis. These are the values of x for which the polynomial function equals zero. In the example polynomial function given, graphing it would allow you to pinpoint these intercepts. They are crucial in understanding the behavior of the function and form the basis for dividing the function's domain into intervals on which the function is either positive or negative.

Real zeros can be determined algebraically or graphically. For the illustrated polynomial, plotting the graph would reveal where it intersects the x-axis. These points represent solutions to the equation that can sometimes be obtained by factoring, synthetic division, or using the Rational Root Theorem. Each real zero corresponds to a factor of the polynomial, so they together can provide insight into the function's factored form.

Nonreal Complex Zeros

Nonreal complex zeros are solutions to polynomial equations that include imaginary numbers. They cannot be observed directly on a real number graph but have a profound impact on the function's nature. For polynomials with real coefficients, the complex zeros appear in conjugate pairs. This means that if a complex number, say 'a + bi', is a zero, then so is its conjugate, 'a - bi'.

In the graphing utility exercise, once the real zeros are located, they can be subtracted from the polynomial's degree to determine the number of complex zeros. For the cubic function given, if one real zero is found, there would be two complex zeros that are conjugates. The Fundamental Theorem of Algebra guarantees that a third-degree polynomial will have three zeros in total when counting both real zeros and nonreal complex zeros.