This lesson introduces students to the Remainder Theorem. A focus of the lesson is that for any polynomial and linear expression , we can state that , where is the remainder and is a polynomial. To get to this endpoint, students begin by considering the meaning of the remainder when dividing with integers, making explicit the connection between dividend, divisor, quotient, and remainder written using long division and written as a multiplication equation.

Next, students identify an unknown coefficient in a polynomial that has a known factor. Students use a consequence of the Remainder Theorem studied previously: If a polynomial has a factor of , then . Using new connections made during their work with division and writing multiplication equations, students now consider this fact from the perspective of what must be true about when we know . This activity is purposely unscaffolded to encourage students to make sense of the problem and choose a solution path (MP1).

In the last activity, students complete repeated calculations to help them make the connection that the value of is equal to the value of the remainder when is divided by (MP8). Once the Remainder Theorem is established, it can then be stated that, for a polynomial , means must be a factor. In previous lessons, students used zeros to predict factors, and now they will know that zeros always correspond to factors in this way.

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making graphing technology available.