Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(y=\frac{1}{4} x^{3}\left(x^{2}-9\right)\)

Step-by-Step Solution

Answer

The zeros of the function \( y=\frac{1}{4} x^{3}\left(x^{2}-9\right) \) found algebraically are x = 0, x = i3, and x = -i3. The graph of the function confirms that the only real zero is x=0, while the other two are imaginary and do not appear on the graph of real numbers.

Step 1: (a) Finding Zeros Algebraically

The expression given is \( y=\frac{1}{4} x^{3}\left(x^{2}-9\right) \). Setting it equal to zero will help find the zeros of this function: \( 0=\frac{1}{4} x^{3}\left(x^{2}-9\right) \). This simplifies to \( 0=x \cdot x^{2} \cdot (x^{2}-9) \). This can be rewritten as \(0=x (x-i\sqrt{9})(x+i\sqrt{9})\), giving the solutions x = 0, x = i3, and x = -i3. These are the zeros of the function found algebraically.

Step 2: (b) Graphing the Function

Using a graphing utility, the function \( y=\frac{1}{4} x^{3}\left(x^{2}-9\right) \) is graphed. Visual representation of the function helps to understand behaviour of the function and to identify approximate zeros.

Step 3: (c) Approximating Zeros Using the Graph

Looking at the graph produced in step (b), the x-intercepts are the zeros of the function. These can be approximated and compared with the zeros found in part (a). The graph intercepts the x-axis at x=0, which is the same zero found algebraically. Additionally, the function does not cross the x-axis at any real numbers other than zero, leading to the conclusion that the other zeros, x = i3, and x = -i3, are imaginary and do not have a graphical representation on the real number plane.

Key Concepts

Zeros of a Polynomial

In algebra, finding the zeros of a polynomial means determining the values of x for which the polynomial equals zero. These are the points where the graph of the polynomial intersects the x-axis. To find these zeros, one typically sets the polynomial equal to zero and solves for x. For the polynomial function given by
\( y=\frac{1}{4} x^{3}(x^{2}-9) \), we have a cubic term and a quadratic term which factor out to
\( 0=x(x^2)(x^2-9) \). By employing the zero product property (if a product equals zero, then at least one of the factors must be zero), the zeros can be found as x = 0 (real zero), and \( x=\pm i\sqrt{9} \) which simplifies to \( x=\pm 3i \) (complex zeros).
These solutions are crucial as they represent the points at which the graph of the polynomial will cross or touch the x-axis. Understanding the nature of these zeros is essential to accurately sketching graphs and solving polynomial equations.

Graphing Utility

A graphing utility is a tool that allows one to visually represent the behavior of functions. For instance, graphing calculators or software like GeoGebra can graph polynomials, trigonometric functions, and more. When it comes to polynomials, a graphing utility paints a picture of the function's curve and displays its intersections with the axes: the x-intercepts being the zeros of the function.
After inputting the function
\( y=\frac{1}{4} x^{3}(x^{2}-9) \), a graphing utility plots a curve that helps us visualize where the zeros are, even if they are not real numbers. The benefit of using such a tool lies in its ability to provide an immediate graphical representation which can often reveal properties and behaviors of the function that are not immediately evident from the equation alone.

Complex Numbers

The concept of complex numbers is a fundamental expansion of our number system comprising both real and imaginary units. A complex number is written in the form a + bi, where a is the real part, b is the coefficient of the imaginary part, and i is the square root of -1.
In the context of finding zeros, when solving polynomial equations, if the discriminant (the part under the square root in the quadratic formula) is negative, one cannot find real number solutions; instead, imaginary or complex solutions emerge. For instance, the equation \( x^2 = -9 \) has solutions \( x=\pm 3i \) which are complex numbers. These solutions are significant in mathematics, even though they do not have a point on the graph where the x-coordinate is a real number.
Complex numbers are used in many fields, including engineering, physics, and mathematics, to solve problems that cannot be addressed with real numbers alone.

Real and Imaginary Zeros

Polynomials can have both real and imaginary zeros. Real zeros correspond to the x-values where the graph of the polynomial crosses the x-axis on the coordinate plane. They are the solutions you can easily spot on the graph because they represent actual points on the plane.
On the other hand, imaginary zeros (also called complex zeros when combined with a real component) are solutions to the polynomial equation that exist in the realm of complex numbers and do not cross the x-axis on a standard graph. For the given function, while we have the real zero at x = 0, the solutions \( x=\pm 3i \) are imaginary. They represent important mathematical concepts but do not show up when graphing the function using only real number coordinates. Understanding the difference between real and imaginary zeros is crucial for a comprehensive grasp of polynomial functions and their graphs.