Relationship between the Degree and the Number of X-intercepts of a Polynomial
Lesson 7 of 9
Objective: Students will generate examples of polynomials to explain why the degree of a polynomial is greater than or equal to the number of x-intercepts of the polynomial.
The warm-up reviews the key skill from the previous day, comparing and contrasting polynomial functions, and provides an organizer for students to start thinking about the relationship between the number of x-intercepts of a polynomial and the degree of the polynomial.
For the first problem, it is a little bit more challenging for students to articulate the affect that this change of the equation has on the graph. The idea mathematically speaking is that raising a factor to the power of 2 changes the multiplicity of that root and this affects the behavior of the function at that root. When a root has a multiplicity of 2, the graph of the polynomial does not cross the x-axis. It “bounces off” the x-axis at this point. During the warm-up work time, I made sure to have this discussion with each group of students. After they had made their graphs and were attempting to explain how the change of the equation affected the graph, I checked in with each group to discuss this. I used the academic vocabulary of root and multiplicity with some groups of students, but with others I used the more informal vocabulary of “bouncing off” depending on how ready students were to deal with the abstract language.
Today’s lesson really hinges on an understanding of the concept of “degree.” I tell my students simply, “it is the highest exponent in the equation when the equation is rewritten without parentheses.” At first, most of my students start by multiplying the binomials using the Distributive Property to actually rewrite the equation without parentheses and then identify the largest exponent. After doing this a few times, students start to realize that they don’t actually have to do this, and start trying to come up with shortcuts. They start to develop the idea that they can just “count the x’s,” which is an informal way of describing a shortcut that will likely work. I ask them to verify their idea by checking their prediction with problems that they actually worked out the long way. This is a perfect example of reason both abstractly and quantitatively and it is worth pointing this out to students explicitly: thinking about the same question in two different ways is a great way to develop a more sophisticated understanding of the question (MP2).
When it comes to the row of the table that is missing a polynomial, I try to circulate and ask each group of students: “Can you write a polynomial that has degree 4 and has 5 x-intercepts?” I try to ask in a voice that doesn’t suggest that it might be impossible. I never use the word impossible nor do I say anything like, “Some of these questions might be impossible.” Some of my students tried really hard to find a polynomial that fit these requirements; other students seemed to promptly decide that it was impossible. No matter the student response is, I ask them, “How could you justify that answer? Are you convinced? How could you convince somebody else?” This really sets up an opportunity for students to try to understand each other’s claims, and the only way to ensure that this happens is if you give no indication of your opinion (MP3).
For the final question, some students say, “There is no relationship between the degree and the x-intercepts,” because they can see from the examples that the degree does not have to be the same as the x-intercept. It is fine if they cannot articulate anything now—I just ask them to write any true sentences that they can come up. Some students simply write: “The degree of the polynomial does not have to be the same as the number of x-intercepts.” Some students claim that: “The number of x-intercepts cannot be greater than the degree of the polynomial.” I just ask all students if they can justify their claims and I give no indication of whether I agree or not.
The exit ticket is important today. During the middle of the lesson, students begin to develop their ideas but they don’t necessarily articulate them. The purpose of the exit ticket is to give them independent think-time to attempt to articulate their ideas. The student responses below demonstrate some understanding and some misconceptions, which are described below.
All three of these sample student responses show clear understanding of the key idea that the number of roots cannot be greater than the degree. They also demonstrate understanding of the idea that you can increase the degree by increasing the multiplicity of a particular root without increasing the number of different roots.
The two questions on this exit ticket that are more difficult for students are the 2nd and the 4th questions. All students below answer that the polynomial must have at least one x-intercept. This makes sense in the context of their prior knowledge because all the graphs they have made so far have had x-intercepts. This misconception is fine for now; a more solid understanding will come forth when we look at horizontal and vertical shifts of even and odd degree polynomials. The 4th question also proved difficult for students, partly because it is not possible so it is harder to explain.