Problem 64 - College Algebra

Question

For the following exercises, 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

The rational zeros of the function are where the graph crosses the x-axis, using a calculator, these can be determined more accurately.

Step 1: Enter the Function into the Calculator

Using your graphing calculator, enter the given polynomial function \(f(x) = 12x^4 + 55x^3 + 12x^2 - 117x + 54\). Ensure that you input the correct coefficients for each term.

Step 2: Graph the Function

Generate the graph of the polynomial function by selecting the appropriate graphing function on your calculator. This will help visualize where the graph crosses the x-axis, which corresponds to the rational zeros.

Step 3: Identify the x-intercepts

Examine the graph to locate the points where the graph crosses the x-axis. These points are the real roots, or rational zeros, of the polynomial.

Step 4: Verify the Rational Zeros

Once you identify the x-intercepts, use the calculator’s zero or root function to accurately determine their values. Typically, these will be whole numbers or simple fractions.

Key Concepts

Graphing Calculators

A graphing calculator is a powerful tool used for visualizing mathematical functions, equations, and inequalities. It allows you to input a polynomial function and quickly produce its graph, which can be incredibly helpful in understanding how the function behaves.

For our example, you'll input the polynomial function given in the exercise: \[ f(x) = 12x^4 + 55x^3 + 12x^2 - 117x + 54 \]

• Start by entering the polynomial into the calculator. Ensure accuracy by carefully checking each coefficient.
• Select the graphing function, and watch as the calculator generates a visual representation of the polynomial.
• This visualization is key to finding rational zeros, as it shows where the curve intersects the x-axis.

Using graphing calculators saves time and reduces the complexity of manual calculations, making it a must-have tool for students dealing with polynomial functions.

Rational Zeros

Rational zeros, or rational roots, are the x-values for which a polynomial equals zero. They can be whole numbers or fractions. In simpler terms, if you plug these x-values into the polynomial, the result will always be zero.

Finding rational zeros of a polynomial is essential because they often represent critical points of the polynomial's graph, such as intersections with the x-axis.

• Use the Rational Root Theorem to understand the potential candidates for rational zeros. This theorem suggests that the possible rational zeros are the factors of the constant term (54 in this case) divided by the factors of the leading coefficient (12 here).
• Make an initial list of potential rational zeros: for example, \( \pm 1, \pm 2, \pm 3, \pm 6 \).
• These possible zeros make it easier to anticipate which points to verify using a graphing calculator or by synthetic division.

Identifying rational zeros is an effective strategy for reducing the complexity of polynomial equations.

X-intercepts

X-intercepts are the points on the graph where the function crosses the x-axis. For polynomial functions, these x-intercepts correspond to the real zeros of the function.

When graphing a polynomial like \[ f(x) = 12x^4 + 55x^3 + 12x^2 - 117x + 54 \]on a calculator, you'll see several points where the curve meets the axis.

• These points are significant because they identify where the function changes direction.
• The x-intercepts are found by looking for where the y-values of the graph read as zero, often clearly marked on graphing calculators.
• Each x-intercept can be verified by testing it as a possible rational zero.

Recognizing these points helps in understanding the behavior and end-behavior of the polynomial function.

Real Roots

Real roots of a polynomial are the values of x for which the polynomial equals zero. In the context of polynomial functions, these roots correspond to x-intercepts and are the solutions to the equation.

For the polynomial given by \[ f(x) = 12x^4 + 55x^3 + 12x^2 - 117x + 54 \],
it is crucial to identify these roots as they have special significance:

• They indicate where the function's output changes the sign.
• Each root represents a point where the curve of the graph touches or crosses the x-axis.
• Real roots are crucial in forming the factorization of the polynomial, aiding in deeper understanding and further calculations.

Finding these roots helps break down the polynomial into more manageable pieces, revealing important characteristics about its graph and potential applications.