Problem 47 - 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). \(y=4 x^{3}-20 x^{2}+25 x\)

Step-by-Step Solution

Answer

The algebraic zeros of the equation \(y=4 x^{3}-20 x^{2}+25 x\) are \(x = 0\) and \(x = 2.5\)

Step 1: Find the zeros algebraically

To find the zeros algebraically, the quadratic equation is factored out as follows: \(y=4 x^{3}-20 x^{2}+25 x = x(4x^2 - 20x + 25) = x(2x - 5)^2\). Setting this to zero, we get the possible values of x as: \(x = 0\) and \(x = 2.5\)

Step 2: Use a graphing utility

Using a graphing tool, input the function \(y=4 x^{3}-20 x^{2}+25 x\) into the tool and plot the graph. You will observe the curve intersecting the x-axis at two points.

Step 3: Approximate zeros from the graph

From the graph, identify the points where the curve intersects the x-axis. These points are the approximate zeros of the function. Compare them with the results from Step 1.

Key Concepts

Finding Zeros

Zeros of a polynomial function are the values of the variable that make the entire function equal to zero. For a function like \(y = 4x^3 - 20x^2 + 25x \), finding zeros involves solving for \(x\) when \(y = 0\).
Set the function equal to zero:

• First, factor out the greatest common factor. Here, that's \(x\), giving us \(x(4x^2 - 20x + 25)\).
• Next, factor the quadratic expression if possible. The quadratic \(4x^2 - 20x + 25\) can be factored as \((2x - 5)^2\).

Putting it all together, we have \(x(2x - 5)^2 = 0\). Solving this:

• For \(x = 0\), the zero is 0.
• For \((2x - 5) = 0\), solve for \(x\) to get \(x = 2.5\).

Thus, the function zeros are \(x = 0\) and \(x = 2.5\). These values tell us where the graph will intersect the x-axis.

Factoring Polynomials

Factoring polynomials involves breaking down a complex expression into simpler components that, when multiplied together, produce the original expression. It's like reverse expanding a product.
To factor \(4x^3 - 20x^2 + 25x\):

• First, identify the greatest common factor, which is \(x\). Factoring \(x\) out gives \(x(4x^2 - 20x + 25)\).
• Next, focus on factoring the quadratic inside: \(4x^2 - 20x + 25\). This can be rewritten as \((2x - 5)^2\), since \( (2x - 5) \times (2x - 5) = 4x^2 - 20x + 25 \).

This factoring breaks our cubic function into \(x(2x - 5)^2\), making it easier to find the zeros or analyze its behavior.

Graphing Utilities

Graphing utilities are tools or software that allow you to visually plot and analyze functions. They can range from graphing calculators to computer programs like Desmos or GeoGebra.
Using a graphing utility, we input the function \(y = 4x^3 - 20x^2 + 25x\) and generate its graph. This visual representation can help you:

• Identify where the graph intersects the x-axis, which are the zeros of the function.
• Understand the shape and behavior of the function, such as whether it opens upwards or downwards, and where it has maximum or minimum points.

The graphical approach complements algebraic methods, offering a visual confirmation of calculated zeros.

Cubic Functions

Cubic functions are polynomial equations of degree three, typically written in the form \(ax^3 + bx^2 + cx + d\). These functions can have up to three real roots (or zeros) and often exhibit S-shaped curves.
In the function \(y = 4x^3 - 20x^2 + 25x\):

• The leading coefficient (4) influences the width and direction of the graph's end behavior. Positive implies that as \(x\) approaches positive or negative infinity, \(y\) does the same.
• The roots we found, \(x = 0\) and \(x = 2.5\), signify the x-intercepts of the graph.
• Such functions might have inflection points where the curve changes concavity. Investigating these helps understand the function's behavior.

Cubic functions play a key role in modeling real-world scenarios where changes don't remain linear, demonstrating how quantities grow, peak, and decline.