SWBAT use the remainder theorem to identify roots of polynomial equations. SWBAT explain the relationship between factors and zeros of a polynomial equation.

Students make sense of the many connections between polynomial factors, zeros, and remainders from polynomial long division.

Today's lesson aims to provide practice doing long division, interpreting the results of long division, using synthetic substitution, and "discovering" the remainder theorem.

For the bulk of the class, students will be working on a series of problems designed to accomplish these goals. I will be circulating among the students offering assistance to groups and individuals as needed.

For advanced classes, I will build in a section of the lesson in which I will help students develop a Proof of the Remainder Theorem in the context of a whole-class discussion of their conjectures. I've included two different proofs in the resources.

5 minutes

As students enter the class, I'll ask them to write on the board the numbers of the problems they need help with. Then, I'll ask for volunteers to put their solutions to these problems up for all to see. All of this should take place while I'm taking attendance and returning any graded work.

My expectation is that the student who had trouble will carefully read through the posted solution, ask the presenter for clarification, and then correct his or her work. If there are any unresolved problems, I'll be able to help the class make sense of them. This shouldn't take more than 10 minutes and everyone should be ready to hand in their homework by the end.

10 minutes

To begin this segment of the lesson, I'll hand out *The Remainder Theorem*. My students will work individually at first so that I can see how they're doing on their own. Also, many of these problems come at the topic from slightly different angles, and I'd like them to have a chance to think it through on their own before discussing it with anyone else. A little bit of perseverance and making sense of problems here (**MP 1**)! Ten minutes of independent work should be enough time for all students to complete the first two problems.

During this time, I will be moving around the room doing some formative assessment. I'll be looking over the students' shoulders to see what sort of progress they're making. I'll pause to ask some questions of individual students now and then; I look for students who are either getting frustrated and making no progress or students who are rushing ahead without taking time to think about what they're doing.

Some students may be confused initially by the first problem, and I think it's okay for them to move ahead to problems 2 or 3 before coming back to it, if they want to. That said, problems 1 through 3 should be totally complete before any student attempts 4 or 5.

10 minutes

20 minutes

Students work quasi-independently to complete problem #3 on The Remainder Theorem. I expect them to be continually checking their results with their peers. If they disagree with a nearby classmate, those two students should briefly work together to resolve their differences. Then, they should continue on their own again. In this way, I expect everyone to finish problem 3 by the end of class and to be reasonably confident that they have the correct solutions.

Since the correctness of these problems is so important to "discovering" the Remainder Theorem, I'll be paying very carefull attention to the students' work. When I see that some student has an incorrect result, I will find a nearby classmate with the *correct* solution and ask those two students to compare their work. This makes it clear that *one of them* is wrong, but they'll have to figure out who (**MP 6 **&** MP 1**). Of course, I'll keep an eye on things to make sure that the correct solution comes out in the end.