SWBAT describe properties of polynomials based on their graphs. SWBAT graph polynomial functions that satisfy specific criteria.

Applying the Remainder Theorem and the Fundamental Theorem of Algebra, students explore the graphs of higher-degree polynomials.

10 minutes

I'll begin this class with a few questions that will prime the pump. Typically, I'll just come right out and ask the class in a straightforward way:

- "Yesterday we proved something called the Remainder Theorem. Can someone tell me what it is? Can someone tell me what it's useful for?"
*It's useful for identifying roots & factors. It gives us the value of the function. It's related to Synthetic Substitution.*- "We've also proven something called the Fundamental Theorem of Algebra recently. Can you tell me about that?"
*It's useful for predicting the number of roots & factors.*

When I'm satisfied with these answers, I'll move on to a quick class example:

**Given a 6th-degree polynomial, what might its graph look like? What is its end behavior? How many times might it cross the x-axis? How many vertices would this produce? Is the y-intercept known?**

Finally, I'll define "parity" as the oddness or evenness of a number and make sure that the class understands that the end behavior of a polynomial depends on the parity of the highest degree term and the sign of its coefficient. For good measure, I'll have some student come to the board to draw the four possible end behaviors on board.

Please see my video for more thoughts on all of this.

20 minutes

At this point, I hand out Higher Degree Polynomials 1 and assign problems 1a - 1d. In this assignment, given graphs of polynomial functions, students interpret the graphs to answer questions about various properties of each function. First, students observe end behavior to identify the parity of the highest-degree term and sign of the leading coefficient. Next, they use the number of vertices and/or roots to determine the *least possible* degree of the polynomial. (Emphasize that the actual degree could be greater but cannot be less.) Finally, students apply their knowledge of the remainder theorem and fundamental theorem to answer more challenging questions requiring some interpretation.

As they work I keep careful track of how students are faring. (See the solutions.) It might be helpful to use a **Think-Pair-Share** strategy for the first problem just to make sure that struggling students don't get left behind. After that, encourage students to discuss the problems as much as they need to in order to understand the solutions (**MP 1**). As I circulate among the students, I make sure to ask as many students as possible to explain the thought process behind their answers. Most of these problems do not require students to provide written explanations, but they should be prepared to offer a verbal explanation for any of their answers! (**MP 3**)

Most questions and misconceptions will be addressed in one-on-one conversations during this section of the lesson, but I still like to take a few minutes to call on the whole class for any lingering questions. Even if some students have not yet completed all parts of problem 1, at the end of 20 minutes I will ask everyone to move ahead to question 2.

15 minutes

Now, I ask my students to move on to Problems 2a - 2f of Higher Degree Polynomials 1. In these problems, students are asked to draw graphs of polynomials functions based not on a given equation but on a description of various characteristics. In each case, students are given the degree of the function (determining end behavior & maximum number of roots/vertices) and a specific number of real roots.

One nice aspect of this exercise is the degree of freedom the students have in drawing their graphs. Since the various graphs may be so different, it is a good exercise to have students check one another's work (**MP 3**). One way to do this is to have students work individually at first. After having a chance to complete most of the graphs, assign students to groups. In groups, students should agree on a single graph for each problem and draw these carefully on a single large sheet of paper. The papers can be hung up for comparison, perhaps in a brief gallery walk at the beginning of the next lesson.

Things to watch for as students draw the graphs are the appropriate end behavior of the function and the number of roots/vertices. Remember, it's okay to have fewer vertices/roots than the degree of the function allows, but it's not okay to have more! In other words, it's okay if a 5th-degree function looks a little like a cubic, but not the other way around. I also watch for "pointy" vertices; students should take the time to draw nice, *smooth* curves.

As class ends, I will assign any unfinished problems for homework.