Question

You are asked to graph the given polynomial. Because the polynomial does not factor, you will have to use the material "Polynomial Functions and their Graphs" to find the zeros. That will give your X intercepts. Combine this with the information on "Zeros of a polynomial Function". Be sure to mark enough points to make the graph as accurate as possible. Be sure to show all work for the following: 1). List all possible zeros and characteristics of them. 2). Show work for testing the zeros until you find all of the zeros ( x intercepts) and their multiplicity. Give the exact value of the x intercepts. Hint: They are not all integers. You will have some irrational zeros. 3). Show your work for finding the y intercept. 4). Show the end behavior and your reasoning for the end behavior. 5). With all of the above work done, you will then be able to mark appropriate points and connect them a good sketch. f(x) = x^4 - 3x³ - 20x² - 24x - 8

Answer 1

Final answer:

To graph the polynomial function, list all possible zeros using the Rational Root Theorem, test these zeros to find the actual ones and their multiplicities, find the y-intercept by evaluating f(0), determine the end behavior based on the leading coefficient, and then plot these points and sketch the curve.

Step-by-step explanation:

Graphing a Polynomial Function

To graph the polynomial function f(x) = x^4 - 3x^3 - 20x^2 - 24x - 8, you'll need to follow several steps involving finding zeros, x-intercepts, the y-intercept, and end behavior.

List all possible zeros: To determine the possible rational zeros, use the Rational Root Theorem, which suggests they are the factors of the constant term divided by the factors of the leading coefficient.

Test the zeros: Substitute the possible zeros into the function to find which are actual zeros. Zeros with a multiplicity greater than one will touch the x-axis and bounce off rather than cross it.

Y-intercept: The y-intercept is found by evaluating f(0), which in this case is -8.

End behavior: Since the leading term is x^4, which is positive, as x approaches infinity or negative infinity, the function will tend to positive infinity. This suggests the ends of the graph face upwards.

After finding the x-intercepts and y-intercept, plot these points on a graph. Connect the points, considering the multiplicity of zeros and end behavior to sketch the curve of the polynomial function accurately.