Problem 438 - Precalculus

Question

Use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational. $$f(x)=12 x^{4}+55 x^{3}+12 x^{2}-117 x+54$$

Step-by-Step Solution

Answer

Rational zeros are approximate x-values where the graph crosses the x-axis.

Step 1: Enter Polynomial Function

First, enter the polynomial function into your graphing calculator. The function is \( f(x) = 12x^4 + 55x^3 + 12x^2 - 117x + 54 \). Ensure all terms are correctly typed.

Step 2: Adjust Viewing Window

Adjust the viewing window on the calculator to make sure you can see the important parts of the graph. A standard window might work, but adjust the x and y ranges until the curve's behavior, especially where it crosses the x-axis, is visible.

Step 3: Graph the Function

Press the graph button to plot the polynomial function. Observe the shape and position of the graph, focusing especially on where the curve crosses the x-axis, as these points indicate real rational zeros.

Step 4: Identify Rational Zeros

Look for x-axis intersections, which represent the zeros of the function. Zoom in if necessary for precise observation. These intersection points are your rational zeros.

Step 5: Confirm Rational Zeros

Use the graphing calculator's root-finding feature or substitute the identified zeros back into the polynomial function to confirm that they indeed satisfy \( f(x) = 0 \).

Key Concepts

Rational Zeros

Rational zeros of a polynomial are the values of x for which the polynomial equals zero, and these zeros can be expressed as a ratio of two integers. In simpler terms, they are the "roots" or "solutions" that satisfy the equation \( f(x) = 0 \). Finding rational zeros is an essential step in solving polynomial equations, especially when dealing with higher-degree polynomials.

• The Rational Root Theorem is a helpful tool here. It states that any possible rational root of a polynomial equation with integer coefficients is a fraction \( \frac{p}{q} \). Here, \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.
• Testing these potential rational zeros can be done using synthetic division or by simply plugging them into the polynomial to check if they result in zero.

Recognizing rational zeros graphically involves identifying x-intercepts on a graph, and using a graphing calculator can expediate the process.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing mathematical functions, especially polynomials. When dealing with polynomial functions, a graphing calculator can help you to quickly plot the curve and visualize its intersections with the x-axis, which correspond to the zeros of the function.

• Beyond its basic graphing capabilities, the graphing calculator has features like zoom, trace, and a grid background that facilitate a better understanding of the graph.
• Inputting the polynomial equation correctly is crucial. Double-check to ensure every coefficient and exponent is entered correctly, as any errors in the input can lead to incorrect visualizations and misinterpretations.

Once the function is graphed, observing the points where the graph crosses the x-axis can guide you towards identifying rational solutions.

Root-Finding Feature

The root-finding feature in a graphing calculator is a specific function designed to precisely identify where a polynomial function becomes zero. This feature can take the guesswork out of figuring out the rational zeros from a graph by providing exact values.

• Once you have graphed the polynomial, access this feature usually through a menu or button labeled as "Calc" or "Root". This allows you to select the regions around the x-intercepts to solve for exact zero values.
• The root-finding tool can automatically calculate and provide the x-values where \( f(x) = 0 \), showing them on the display for confirmation.

Using this feature ensures that the rational zeros identified visually or hypothesized are indeed accurate and satisfy the polynomial equation.

Viewing Window Adjustment

Adjusting the viewing window on your graphing calculator is vital for accurately displaying the graph of your polynomial function. Without proper adjustment, important features of the graph, such as x-intercepts, may be missed or not visible.

• The viewing window sets the range for the x and y axes. Adjust the x-axis values to cover the domain of interest and the y-axis values to see the full behavior of the graph, particularly where it crosses or approaches the x-axis.
• If the standard view doesn't display the critical parts of the graph, manually adjust the window settings. Aim to include the zero crossings within the window range while keeping the graph centered and scaled.
• Effective viewing window adjustments often involve trial and error. After each adjustment, re-graph the function to ensure a clear view of all potential x-intercepts.

By methodically refining the window, you enhance the visibility of the rational zeros and make the exploration more efficient.