Problem 80 - Algebra and Trigonometry

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. $$f(x)=3 x^{5}-2 x^{4}+6 x^{3}-4 x^{2}-24 x+16$$

Step-by-Step Solution

Answer

Upon graphing the polynomial equation, all instances where it intersects the x-axis reveal the quantity of real zeros for the polynomial. To determine the number of imaginary zeros, subtract the number of real zeros from the degree of the polynomial, in this case 5. As an example, if the polynomial has 2 intersection points hence 2 real zeros, it will have 5-2=3 imaginary zeros.

Step 1: Graphing the Polynomial Function

To start with, a graphing utility should be used to sketch the function. The utility should be used to plot \(f(x) = 3x^5 - 2x^4 + 6x^3 - 4x^2 - 24x + 16\). The x-intercepts on the graph represent the real zeros of the function. These are the points where the function intersects the x-axis.

Step 2: Identifying Real Zeros

Once the graph has been created, the places where the graph intersects the x-axis need to be identified. These intersections represent the real zeros of the function. Count these to find the total number of real zeros.

Step 3: Determining Imaginary Zeros

To find the number of imaginary zeros (complex roots), use the Fundamental Theorem of Algebra, which states that a polynomial of degree n has exactly n complex roots. If the degree of the polynomial is 5 (as it is here), then the total number of zeros, both real and imaginary, is 5. Subtract the number of real zeros from 5 to get the number of imaginary zeros.

Key Concepts

Graphing Utility

The use of a graphing utility is a powerful technique for visualizing polynomial functions. By plotting a function, such as \(f(x) = 3x^5 - 2x^4 + 6x^3 - 4x^2 - 24x + 16\), on a graph, the student can immediately see the behavior of the polynomial in a visual format. This includes where it intersects the x-axis, which indicates the real zeros, and how the curve moves, showing intervals of increase or decrease.

Graphing utilities are commonly available in a variety of formats, from online graphing calculators to specialized software. To effectively use a graphing utility, input the polynomial equation, ensure the proper scale is set to capture all relevant features, and look for the x-intercepts. These are key in understanding the nature of the polynomial's zeros and can provide insight into many other properties of the function.

Real Zeros

Identifying the real zeros of a polynomial is essential because they are the solutions to the equation where the polynomial equals zero. In mathematical terms, if \( f(x) = 0 \), then x represents a real zero of the polynomial function.

For the given function \(f(x)\), real zeros correspond to the x-values where the graph crosses the x-axis. These zeros can be represented as points on the real number line. Once the graph is created using a graphing utility, the x-intercepts can be determined visually or using features within the utility that may precisely calculate these intercepts. It’s important for students to understand that not all zeros of a polynomial need to be real; some may be imaginary or complex.

Imaginary Zeros

While real zeros can be visualized on a graph, imaginary zeros cannot be observed directly from the x-axis intersections. These zeros arise when dealing with complex numbers and typically occur in conjugate pairs. For instance, if \(a + bi\) (where i is the imaginary unit) is a zero, then its conjugate \(a - bi\) is also a zero of the polynomial.

In the context of the exercise, students will not find imaginary zeros plotted on the real plane graph. Instead, imaginary zeros are deduced mathematically after identifying all the real zeros. If a polynomial of degree n has fewer than n real zeros, the remaining zeros will be imaginary. These complex numbers are a fundamental part of understanding polynomials, especially when dealing with higher degrees and non-real solutions.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-zero single-variable polynomial with complex coefficients has at least one complex root. This theorem further implies that a polynomial of degree n will have exactly n roots, when counted with multiplicity. These roots may be real or non-real (imaginary).

This means that for any polynomial given in the exercise, such as a quintic polynomial (which has a degree of 5), there will be a total of five zeros. After identifying the real zeros with a graphing utility, the remaining zeros, if not visible on the graph, will be imaginary. This theorem is a guarantee of the number of solutions and plays a critical role in solving polynomial equations analytically and understanding their full set of roots.