The purpose of this Warm-up is to elicit the idea that we can use long division to divide polynomials similarly to how we have used long division to divide numbers, which will be useful when students do polynomial long division in a later activity. While students may notice and wonder many things about these images, relationships between the division steps for the integers and the division steps for the polynomials are the important discussion points.

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

Then display the completed long division for all to see:

Tell students that they are going to learn another way to do polynomial division. Make sure to note the 0 remainder with this example. Just as getting the same constant term in the last entry in the diagrams used in the previous lesson meant that the polynomial was divided evenly, getting a remainder of 0 in polynomial long division means the divisor is a factor of the dividend. This is similar to how we know that 11 is a factor of 2772 because .

In this activity students make use of structure (MP7) as they make connections between polynomial long division and the reasoning used in completing a diagram. They also critique a statement that is intentionally unclear and incorrect and improve it by clarifying meaning, correcting errors, and adding details (MP3).

The remainders for division in this activity continue to be 0 since the focus is on dividing by known linear factors. In future lessons, students will focus on what a non-zero remainder means both for polynomials and for rewriting expressions of rational functions.

Students practice using long division to factor when one factor is already known. Both polynomials in this activity were factored by students in a previous lesson using a diagram, and students will compare the two methods during the whole-class discussion.

The first question has the division started, to give students an example to study.  Students are working through the logic of the polynomial division to identify other factors, so graphing technology is not an appropriate tool.

This activity provides additional practice with factoring and dividing polynomials. Earlier questions focus on factoring. Later questions can be solved by dividing by known factors. Not all questions need to be used, depending on the needs of your students.

Monitor for students who use efficient methods to calculate the unknown values. For example, students may find a missing constant term of a factor by using the fact that multiplying all the constant terms of the factors will yield the constant term of the standard-form expression.