Unit 2, Lesson 12
Polynomial Division (Part 1)
Algebra 2
12.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSA-APR.B.2
Know and apply the Remainder Theorem: For a polynomial and a number , the remainder on division by is , so if and only if is a factor of .
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this Warm-up is to elicit the idea that diagrams are a useful way to stay organized when multiplying and dividing polynomials, which will be useful when students multiply and divide polynomials in a later activity. While students may notice and wonder many things about these equations and diagrams, the relationships between the entries in the diagram and the equations are the important discussion points.
Launch
Display the equations and diagrams 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.
Activity
NoneWhat do you notice? What do you wonder?
A.
| 5 | ||
|---|---|---|
| -3 | -15 |
B.
| -4 | |||
|---|---|---|---|
| -1 | +4 |
C.
| -2 |
|---|
Student Response
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Building on Student Thinking
Activity Synthesis
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 diagram or equation. 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.
If finishing the last diagram does not come up during the conversation, ask students to discuss how they could do so starting with the entry above , and why it must be .
12.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
The goal of this activity is for students to understand how a diagram can be used to organize dividing polynomials. This activity continues an idea started earlier—if is a zero of a polynomial function, is a factor of the expression?
After students work out what quadratic times is equivalent to the original expression. They factor the quadratic using a method of their choosing and then make a sketch of the 3rd-degree polynomial.
12.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
Students practice using a diagram to factor when one factor (or more) is already known. The first few questions have diagrams started to give students more examples to study (in particular, that 0 is an acceptable entry). Students use repeated reasoning (MP8) as they work through the logic of polynomial division to identify other factors, so technology is not an appropriate tool.
Launch
Arrange students in groups of 2. Tell students to complete a diagram and then to discuss their thinking with their partner. If necessary, work to reach an agreement before moving on to the next question. Remind students that are writing their result as a product of linear factors, so finishing the diagram does not finish the question.
MLR8 Discussion Supports. Display sentence frames to support partner discussion. Examples:
- “Why did you ?”
- “I agree/disagree because .”
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Present functions 1–3 first. Then check in with students to provide feedback and encouragement before presenting functions 4 and 5. Consider recommending a structured turn-taking to students as they divide by each factor to ensure they start with the result of the previous division each time.
Supports accessibility for: Attention, Social-Emotional Functioning
Supports accessibility for: Attention, Social-Emotional Functioning
Lesson Synthesis
The goal of this discussion is to consider different ways to organize polynomial division. If a linear factor of a polynomial is known, a diagram is one organization strategy to identify other factors, though many other organization strategies are possible. Display the following for all to see:
The polynomial has a known linear factor of . Rewrite the quadratic as two linear factors.
Ask students how they can factor if they don’t use the diagram, and invite students to share strategies they could use, recording them for all to see. For example, students might reason that the other factor must be because and . If time allows, have students work through multiple strategies for factoring the quadratic.
Student Lesson Summary
What are some things that could be true about the polynomial function defined by if we know ?
- Thinking about the graph of the polynomial, the point must be on the graph as a horizontal intercept.
- Thinking about the expression written in factored form, could be one of the factors, since when .
How can we figure out whether actually is a factor?
If we assume that is a factor, then there is some other polynomial where , , and are real numbers and . In the past we have expanded to find . Instead, we can work out the values of , , and by thinking through the calculation backward.
One way to organize our thinking is to use a diagram. First, fill in and the leading term of , . From this we can see the leading term of must be , meaning , since .
We can fill in the rest of the diagram using similar thinking and paying close attention to the signs of each term. For example, we put in a in the bottom left cell because that’s the product of and . But that means we need to have a in the middle cell of the middle row, since that’s the only other place we will get an term, and we need to get once all the terms are collected. Continuing in this way, we get the completed table:
| +24 |
Collecting all the terms in the interior of the diagram, we see that , so . Notice that the 24 in the bottom right was exactly what we needed, and it is how we know that is a factor of . With a bit more factoring, we can say that .