Lesson 13

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. 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.

Display the long division images for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Tell students that today, they are going to learn another way to do division. Make sure to note the 0 remainder with this division. 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 \(2772=(11)(252)\).

The purpose of this activity is for students to make connections between the polynomial division reasoning they did in the previous lesson and polynomial long division. The remainders for division in this activity continue to be zero since the focus is on dividing by known linear factors. In future lessons, students will focus on what a non-zero remainder means for both polynomials and when rewriting expressions of rational functions.

Identify students who make errors while working out the long division (for example, adding a term they meant to subtract) who are willing to share what they did during the discussion.

After 5 minutes of work time, pause the class and display the long division and the incomplete diagram in the first question for all to see. Select students to explain how they completed the diagram and any connections they see between terms in the long division and the diagram. Remind students that since they are dividing by known linear factors, the results of any long division should have a remainder of 0.

Some students may forget that they are subtracting the terms in each new row of long division. Prompt these students to consider the signs in front of the terms they write carefully.

Since polynomial long division can prove challenging due to errors in arithmetic, such as adding instead of subtracting terms, the focus of this discussion is on errors. Invite students previously identified to share the error they made and how they revised their work.

Students have now seen two ways of representing division: using a diagram to work backwards to determine what to multiply the divisor by, and using long division to find the quotient directly. This activity asks students to translate between the two methods so that they see how they are related. Students should understand that these are two strategies for doing the same thing, and that they each have advantages.

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

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

Arrange students in groups of 2. Tell partners to complete the long division individually and then check in with one another to make sure they agree on the terms. If partners do not agree, they should work to reach agreement before moving on to the next question. Remind students that they are asked to write their result as a product of linear factors, so finishing the long division does not finish the question.

Invite 1–2 students per question to share their long division work and how they calculated the other linear factors of the original expression. If not pointed out by students, highlight how, as with diagrams, the order of division does not matter for the final result.

Conclude the discussion by displaying for one of the polynomials, \(A\) or \(C\), the way that students factored it in a previous lesson using diagrams next to the way they factored it using long division in this activity. Give students brief quiet think time to consider what is similar and what is different about the two methods, and then invite students to share their observations.

This activity is optional practice with factoring and dividing polynomials. Not all of the questions need to be used, depending on the needs of your students. The earlier questions are more helpful for students who need practice factoring. The later questions can be solved by dividing by known factors, so they are more appropriate for students who need practice with this skill.

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.

Arrange students in groups of 2–4. After 3–5 minutes of quiet work time, ask students to compare their responses with their groups, taking time to explain their thinking or ask clarifying questions when they disagree on the solutions. Follow with a whole-class discussion.

Invite 1–2 students per question to share their reasoning. Select previously identified students to share unique strategies where possible.