The Remainder Theorem

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SWBAT demonstrate that substituting a value, c, into a polynomial expression yields the same value as the remainder when the polynomial is divided by x-c.

Big Idea

Any polynomial p(x) can be written as a product of (x – a) and some quotient q(x), plus the remainder p(a).

Warm-up and Homework Check

20 minutes

While I check homework with the homework rubric, my students work on Warm-up Remainder Theorem, which is a worksheet I created to help students discover the pattern expressed in the Remainder Theorem.  Students are given pairs of polynomials and integer values. They are instructed to first substitute the value in for x in the polynomial and then divide out the factor associated with that value.   By completing several of these problems, they eventually discover that P(c) is the same value as the remainder when P(x) is divided by (x-c).  In this way, students are introduced to the Remainder Theorem, on which we will take notes later in the lesson [MP8].

Structured Discussion about Remainder Theorem

30 minutes

When my students have had the opportunity to work through the Warm Up, I ask them to discuss the pattern that emerged with the people at their table.  While students work together, I circulate to determine the order in which the tables will present their ideas. My goal for this discussion is to help my students develop a strong conceptual understanding of the idea expressed in the Remainder Theorem.  For this reason, I structure the discussion so that the core idea is introduced first and we build towards the formal mathematical notation [MP3]. To keep the conversation moving in this direction, I use phrases like

  • Can you build on what <Student> has shared?
  • Is there any part of what <Student> has said that you disagree with? 
  • Can you express that idea more formally?
  • Do you see a connection between ___ and ___?

When I am satisfied that my students have a good conceptual understanding of the Remainder Theorem, we take some formal notes with examples.  Examples focus on using the Remainder theorem to identify one root of a polynomial, divide it out of the polynomial expression and then factor the remaining polynomial.  The guided practice that follows the discussion will provide additional reinforcement on this process.

Although many teachers take time at this point in the curriculum to teach students synthetic division and synthetic substitution I do not.  Although this algorithm saves students a bit of time in performing long division of specialized cases, it is not specifically called for in the CCSS. Furthermore, students do not have the mathematical background to understand why it works and therefore think of it as a process to memorize.  I feel time is much better spent in developing the conceptual understanding of the Remainder Theorem.  


Practice with Remainder Theorem

15 minutes

My students will practice using the Remainder Theorem by completing WS Solving Polynomials with the Remainder Theorem with their table partners.  This worksheet contains 4 challenging problems in which students use the Remainder Theorem to identify a root, divide it out using long division, and then find the zeros of the quotient polynomial [MP1]. 

Exit Ticket and Discussion of the Upcoming Quiz

15 minutes

As an exit ticket, I ask students to close their notebook and see if they can state the Remainder Theorem on an index card which they will hand to me.  I tell my students that it is not necessary to memorize specific wording for this theorem, and that what I really want to know is if they understand the idea conveyed by the theorem [MP6]. 

I remind students that they will have a quiz tomorrow on factoring and solving polynomial equations.  To be successful on this quiz, they will need to know how to 

  • factor higher order polynomials
  • divide one polynomial by another
  • use the remainder theorem to identify a root of a polynomial
  • use the Zero Product Property

The homework for the evening is to review notes and assignments in order to prepare for this quiz.