Problem 82 - 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. $$ f(x)=x^{6}-64 $$

Step-by-Step Solution

Answer

The function \(f(x) = x^{6}-64\) has 6 real zeros, at x=-2 and x=2 (each of multiplicity 1), and no imaginary zeros.

Step 1: Graphing the Function

Firstly, graph the function \(f(x)=x^{6}-64\) using a graphing calculator or software.

Step 2: Identification of Real Zeros

Find the points at which the graph intersects the x-axis. These are the real zeros of the function. Looking at the graph, it should intersect the x-axis at x=-2 and x=2. Therefore, there are 6 real zeros - each of multiplicity 1.

Step 3: Identification of Imaginary Zeros

As a sixth degree polynomial with 6 real roots, there cannot be any imaginary roots. The Fundamental Theorem of Algebra states that the number of zeros (real or imaginary) cannot exceed the degree of the polynomial, in this case 6.

Key Concepts

Real Zeros

Real zeros are values of \(x\) where the polynomial equals zero. Think of these as places where the graph of the function touches or crosses the x-axis. For the polynomial \(f(x) = x^6 - 64\), identifying real zeros is crucial. We can first consider the factorization of the polynomial. Rewriting, we have \(f(x) = (x^3 - 8)(x^3 + 8)\) using the difference of squares method.
Now, within each of these factors, we can factor further if possible using methods like synthetic division or polynomial division, but for simplicity, identifying directly from the structure \((a-b)(a+b)=a^2-b^2\), just look for zeroes for solutions like \(x^3 = 8\) and \(x^3 = -8\).
Finally, calculate these values by solving the equations \(x^3 = 8\) which gives \(x = 2\) and \(x^3 = -8\) which gives \(x = -2\). Thus, the real zeros of our polynomial are \(x = -2\) and \(x = 2\).

Imaginary Zeros

Imaginary zeros occur when the solutions to the polynomial involve the square root of a negative number, represented usually in terms of \(i\), where \(i^2 = -1\). These are the zeros you don't see on the graph, as they don't cross the x-axis.
However, for the polynomial \(f(x) = x^6 - 64\) given in the exercise, all zeros are real. This is because the polynomial resolves neatly to real number factors and roots with no negative square roots involved. The absence of imaginary zeros fits logically with the observation that the degree aligns with having only six real zeros, and none imaginary, as expected of a sixth-degree polynomial.
In other cases, it’s crucial to remember that a polynomial’s degree dictates the possible total number of zeros, real or imaginary, that could exist according to the Fundamental Theorem of Algebra.

Graphing Utilities

Graphing utilities are fantastic tools for visualizing polynomial functions. Tools like graphing calculators or software such as Desmos make complex graphs easier to understand by depicting changes over intervals. With these tools, you can quickly graph the expression \(f(x) = x^6 - 64\), and immediately see where the curve intersects the x-axis.
When you use a graphing utility, you get a visual reference to help identify real zeros, recognize symmetry, and understand the overall shape of the graph. For instance, the graph shows us that it crosses the x-axis at points \(x = 2\) and \(x = -2\).
For the sixth-degree polynomial in the problem, seeing these intersections confirms the real zeros calculated algebraically and reassures that there are no imaginary components visible on this two-dimensional plane.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra provides an essential insight into the behavior of polynomial functions. It guarantees that every non-constant polynomial function of degree \(n\) will have exactly \(n\) roots, although some may be repeated and others may be non-real (imaginary).
For the polynomial \(f(x) = x^6 - 64\), it is a sixth-degree polynomial, meaning it will have exactly six zeros. These zeros include both real and any potential imaginary ones. However, as previously identified, this specific polynomial has six real zeros \(x = 2\) and \(x = -2\) each counted with multiplicity.
Understanding this theorem assists in anticipating and validating the total number of solutions a polynomial might possess, ensuring no solutions are missed or miscounted during calculation or graph analysis.