BUILDING POLYNOMIAL FUNCTIONS
Created by Trey Cox, 2014, using http://tedcoe.com/math/
Objectives:
SWBAT explain the relationship between linear factors of a polynomial function and the graph of the function.
SWBAT find the equations of lines that could be components of the polynomial, given the graph of a
polynomial,
Description:
This lesson includes an initial interactive activity for discovering polynomial zeros, an extension into
polynomial parameters, and teacher implementation notes. The primary focus is for students to see the
connections between different polynomial families and the number of roots is demonstrated using interactive
applets in Geogebra (http://tedcoe.com/math/algebra/roots-and-zeros).
Activity:
1. Begin by graphing a polynomial function with 1 factor (make numfactors = 1), 2 factors (make numfactors =
2), 3 factors (make numfactors = 3), 4 factors (make numfactors = 4), 5 factors (make numfactors = 5), and 6
factors (make numfactors = 6). As you do this, also change the constant values within each factor by using the
colored sliders (read, orange, yellow, green, etc.) Be aware of the number of zeros you are getting and then fill
in the table below.
Number of factors in
polynomial function
1
2
3
4
5
6
Number of zeros/xintercepts
2. You should have noticed that there isn’t only one answer for each cell in the table above. Why is this?
Explain what happened to the number of zeros as you changed the constants in each linear factor for the
polynomial. If you need help try a polynomial with 4 factors and then make 2 of the constants in the linear
factors the same number and describe what happens. Then make 3 of the constants the same. Use this
information to help you explain.
Trey Cox, Chandler-Gilbert Community College, 2014. trey.cox@cgc.edu. This
work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
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3. Using the applet is there a way you can fill out the table below?
Number of factors in
MAXIMUM number of
polynomial function
zeros/x-intercepts
1
2
3
4
5
6
4. Consider the data you collected and explain why the number of zeros comes out to the values that they do?
In other words, what is it about the number of factors that impacts the number of zeros?
5. Explain how the constants in the linear factors are related to the zeros of the function. Go to the applet and
use the polynomial function with 3 factors and constants 3, 2, and 1 to explain. Below is a screenshot of the
function so you know how to set up the applet.
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6. The end behavior of polynomial function refers to what the function values behave as the input values get
very large (positive) or very small (negative). Explain how one can use the input slider to show why the
polynomials with an odd number of factors behaves similarly and why those polynomials with an even number
do too. Refer the output values of the function as x ® ¥ and x ® -¥ .
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