Question

Use the Rational Zero Test to list all possible rational zeros of \(f\). Then use a graphing utility to graph the function. Use the graph to help determine which of the possible rational zeros are actual zeros of the function. $$f(x)=2 x^{4}-x^{2}-6$$

Short answer

First, apply the Rational Zero Test which gives possible zeros as \(\pm1, \pm2, \pm3, \pm6, \pm1/2, \pm3/2\). Then, by analyzing the graph of the function, you can confirm which of these are actual zeros of the function.

Step-by-step solution

Apply Rational Zero Test

The Rational Zero Test says that any possible rational zero of a polynomial must be a factor of the constant term divided by a factor of the leading coefficient. In this case, the constant term is -6 and the leading coefficient is 2. Hence, the possible rational zeros are \(\pm1, \pm2, \pm3, \pm6, \pm1/2, \pm3/2\).

Use a Graphing Utility

To determine which of these possible zeros are actual zeros of the function, plot the function using a graphing utility. Look for points on the x-axis where the function intersects, as these represent the zeros of the function.

Identifying the Zeros

After graphically analysing the function, identify the points where the curve intersects the x-axis. Those points are the actual zeros of the function.

Key concepts

Polynomial Zeros

The zeros of a polynomial are the values of the variable where the polynomial evaluates to zero. These zeros can also be called solutions or roots of the polynomial equation. Finding the zeros is crucial as it helps us understand the behavior of the polynomial function graphed in the coordinate plane.
For any polynomial function, zeros are points where the graph intersects the x-axis. If you substitute these values into the polynomial equation, the result should be zero. In our example, with the polynomial function \( f(x) = 2x^4 - x^2 - 6 \), we use methods such as the Rational Zero Test to estimate where these zeros may occur. Once identified, these reasonable guesses about zeros can be verified by actual graphing or further algebraic exploration.

Graphing Utility

A graphing utility, such as a graphing calculator or software, is a powerful tool for visualizing polynomial functions. These tools plot the function on a coordinate grid, making it easier to spot where the function crosses the x-axis, which indicates the zeros of the polynomial.
In the given exercise, after applying the Rational Zero Test, we obtain a list of potential zeros. By inputting the function \( f(x) = 2x^4 - x^2 - 6 \) into a graphing utility, we can see which of these potential zeros are actual zeros by identifying the intersection points on the x-axis.
Graphing utilities not only provide a visual confirmation but also allow for a deeper exploration of the function's behavior, such as identifying turning points and understanding the overall shape of the graph.

Constant Term

In a polynomial, the constant term is the term that does not contain any variable. It offers valuable insights concerning the polynomial's roots when applying certain tests or theorems like the Rational Zero Test.
For the polynomial \( f(x) = 2x^4 - x^2 - 6 \), the constant term is \(-6\). This number is integral when finding potential rational zeros. We pair it with factors of the leading coefficient to list all possible rational zeros.
Understanding the role of the constant term in the formula helps in structuring our search for roots. It is essential to consider both the sign and the magnitude of this term when listing possible rational zeros.

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. This coefficient significantly impacts the graph's shape and the nature of the polynomial's roots.
In the polynomial \( f(x) = 2x^4 - x^2 - 6 \), the leading coefficient is \(2\). We use it, alongside the constant term, in the Rational Zero Test to find potential rational zeros. Our list of possible zeros includes both positive and negative factors of the constant term divided by factors of the leading coefficient, resulting in possible zeros such as \( \pm 1, \pm 2, \pm 1/2\) and so on.
The leading coefficient also provides clues about the end behavior of the polynomial’s graph; for example, a positive leading coefficient in a fourth-degree polynomial typically means that both ends of the graph go upwards.