Problem 59 - Precalculus : Building Concepts and Connections

Question

Graph the function using a graphing utility, and find its zeros. $$p(x)=x^{3}+(3+\sqrt{2}) x^{2}+4 x+6.7$$

Step-by-Step Solution

Answer

The zeros of the function can only accurately be determined with a graphing utility due to the complexity of the coefficients. They are the x-values where the function intersects the x-axis.

Step 1: Understand the Function

The function \(p(x)=x^{3}+(3+\sqrt{2}) x^{2}+4 x+6.7\) is a cubic function, which typically has a distinct 's-shaped' curve. The leading term is positive, indicating that the function opens up. The degree of the polynomial is 3, which suggests that it can have up to 3 real roots.

Step 2: Graph the Function

Use a graphing utility to generate a graph for the function. While it can be done manually, a graphing utility is a more efficient way to visualize the function. To graph, the function is inserted into the graphing tool, ensuring to insert the correct coefficients for each term.

Step 3: Find the Zeros

The zeros of a function are where it intersects the x-axis, or where \(p(x) = 0\). They can be found by observing the x-values of where the graph touches the x-axis. The graphing utility may have an option that finds these zeros automatically.

Key Concepts

Understanding Graphing Utility

Graphing utilities are tools that help visualize mathematical functions. They allow us to see the overall behavior of a function, including how it changes and where it crosses the axes. These utilities can be online platforms or devices like graphing calculators. They are especially useful for complex functions, such as cubic functions, which may not be easy to graph manually.

To use a graphing utility:

• Input the function by entering the equation. Make sure to accurately input each coefficient to ensure precise results. In our case, the function is given as \(p(x) = x^{3} + (3+\sqrt{2}) x^{2} + 4x + 6.7\).
• Adjust the window settings if necessary to see the relevant parts of the function. The visible range should show where the function's curve is crossing the x-axis.

Using a graphing utility is much faster and can provide a clearer picture compared to plotting manually, giving insights into the nature of the function quickly.

Exploring Real Roots

Real roots of a function are the x-values where the function equals zero. For cubic functions like the one in our problem, we can have up to three real roots.

With the help of a graphing utility, real roots are easily identified as the points where the graph crosses the x-axis. Understanding the number and nature of these roots gives us insight into the behavior of the function.

For example, a cubic function might have:

• Three distinct real roots: The graph crosses the x-axis at three separate points.
• One real root and a pair of complex conjugate roots: The graph crosses the x-axis only once, and the other roots do not appear visually.

In our function \(p(x) = x^{3} + (3+\sqrt{2}) x^{2} + 4x + 6.7\), using a graphing utility helps determine the exact x-values of these real roots.

Zeros of a Function Explained

The zeros of a function are specific values for \(x\) that make the function value zero, or \(p(x) = 0\). These correspond to the x-intercepts of the graph. Identifying the zeros is crucial because it directly relates to solving equations.

When graphing a function, you can easily spot the zeros as the points where the graph hits the x-axis. This visual method is a quick way to find solutions that might be cumbersome algebraically. Many graphing tools have built-in features to find zeros automatically, saving time and effort.

Beyond their role in solving equations, zeros also provide insight into the function's characteristics. In a cubic function like \(p(x)\), zeros tell us where the polynomial changes sign, showing us crucial aspects of the function's "personality."

Using the graphing utility for our problem, observe where the plot touches the x-axis to find these zeros efficiently.