Lesson 13

Polynomial Division (Part 2)

13.1: Notice and Wonder: Different Divisions (10 minutes)

Warm-up

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.

Launch

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.

Student Facing

What do you notice? What do you wonder?

Long division for 2772 divided by 11.
4 rows.
Top row: 2.
Second row: 11 on the left, long division symbol, 2,772 inside.
Third row: 22.
Horizontal line.
Fourth row: 5.
 
Long division.
Long division.

\(\displaystyle \displaystyle \displaystyle \require{enclose} \begin{array}{r}   2x^2\phantom{+7x+22} \\  x+1 \enclose{longdiv}{2x^3+7x^2+7x+2} \\     \underline{\text-2x^3-2x^2} \phantom{+7x+22} \\   5x^2+7x \phantom{+22} \end{array} \)

Student Response

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Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

After students share out, display for all to see the completed long division:

\(\displaystyle \displaystyle \displaystyle \require{enclose} \begin{array}{r}   2x^2+5x+2 \\  x+1 \enclose{longdiv}{2x^3+7x^2+7x+2} \\     \underline{\text-2x^3-2x^2} \phantom{+7x+22} \\   5x^2+7x \phantom{+22} \\ \underline{\text-5x^2-5x} \phantom{+22} \\ 2x+2 \\ \underline{\text-2x-2} \\ 0 \end{array} \)

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 \(2772=(11)(252)\).

13.2: Polynomial Long Division (15 minutes)

Activity

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.

Launch

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.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation, such as: “First, I _____ because . . . .”, “Next, I . . . .”, “I tried _____ , and what happened was . . . .”, and “How did you get . . . ?” Encourage students to become error detectives and positively reinforce when students are able to explain their mistakes.
Supports accessibility for: Language; Social-emotional skills

Student Facing

  1. Diego used the long division shown here to figure out that \(6x^2-7x-5 = (2x+1)(3x-5)\). Show what it would look like if he had used a diagram.

    \(\displaystyle  \displaystyle \require{enclose} \begin{array}{r}   3x-5 \\   2x+1 \enclose{longdiv}{6x^2 - 7x - 5} \\      \underline{\text-6x^2 - 3x} \phantom{-55} \\    \text-10x-5 \\  \underline{10x+5} \\    0  \end{array} \)

                                 
    \(2x\) \(6x^2\)  
    1    


    Pause here for a whole-class discussion.

  2. \((x-2)\) is a factor of \(2x^3 - 7x^2 + x + 10\), which means there is some other factor \(A\) where \(2x^3 - 7x^2 + x + 10=(x-2)(A)\). Finish the division started here to find the value of \(A\).

    \(\displaystyle \displaystyle \require{enclose} \begin{array}{r}   2x^2 \phantom{+x+100} \\   x-2 \enclose{longdiv}{2x^3-7x^2+x+10} \\      \underline{\text-2x^3+4x^2} \phantom{+x+100} \end{array}\)

  3. Jada used the diagram shown here to figure out that \(2x^3 + 13x^2 + 16x + 5 = (2x+1)(x^2+6x+5)\). Show what it would look like if she had used long division. 
     

      \(x^2\) \(6x\) 5
    \(2x\) \(2x^3\) \(12x^2\) \(10x\)
    1 \(x^2\) \(6x\) 5

    \(\displaystyle \displaystyle \require{enclose} \begin{array}{r}    2x+1 \enclose{longdiv}{2x^3+13x^2+16x+5} \end{array}\)

Student Response

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Student Facing

Are you ready for more?

  1. What is \((x^4-1) \div (x-1)\)?
  2. Use your response to predict what \((x^7-1) \div (x-1)\) is, and then use division to check your prediction.

Student Response

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Anticipated Misconceptions

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.

Activity Synthesis

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.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Use this routine to give students the opportunity to discuss a common error that arises when doing long division of polynomials. For example, “In Jada’s problem, I placed the term “\(x\)2” under “\(13x\)2” because I multiplied and got the answer.” Ask students to identify the error, critique the reasoning, and write an improved explanation. As students discuss with a partner, listen for students who identify and clarify the incorrect and ambiguous language in the statement. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to explain why subtracting is the appropriate operation when combining these terms. This helps students evaluate, and improve on, the written mathematical arguments of others, as they discuss common arithmetic errors in long division.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

13.3: More Long Division (10 minutes)

Activity

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.

Launch

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.

Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, scaffold partner checks by making them more frequent to support metacognition and error detection. Ask students to complete partner checks after each step rather than at the end of their work.
Supports accessibility for: Memory; Organization

Student Facing

Here are some polynomial functions with known factors. Rewrite each polynomial as a product of linear factors using long division.

  1. \(A(x)=x^3 - 7x^2 - 16x + 112\), \((x-7)\)
    \(\displaystyle  \displaystyle \require{enclose} \begin{array}{r}   x^2\phantom{55555555555} \\   x-7 \enclose{longdiv}{x^3 - 7x^2 -16x+112} \\       \underline{\text-x^3 + 7x^2} \phantom{.55555555555} \\  \end{array} \)
  2. \(C(x)=x^3 - 3x^2 - 13x + 15\), \((x+3)\)

Student Response

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Activity Synthesis

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.

Conversing: MLR2 Collect and Display. During the discussion, listen for and collect the language students use to describe the similarities and differences between using diagrams and using long division. Call students’ attention to language referring to the placement of specific terms, advantages and disadvantages of each method, and words such as “quotient,” “divisor,” and “factors.” Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students compare and contrast the two methods for dividing polynomials.
Design Principle(s): Maximize meta-awareness

13.4: Missing Numbers (15 minutes)

Optional activity

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.

Launch

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.

Student Facing

Here are pairs of equivalent expressions, one in standard form and the other in factored form. Find the missing numbers.

  1. \(x^2 + 9x + 14\) and \((x+2)(x+\boxed{\phantom{30}})\)
  2. \(x^2 - 9x + 20\) and \((x-\boxed{\phantom{30}})(x-\boxed{\phantom{30}})\)
  3. \(2x^2+2x-24\) and \(2(x+\boxed{\phantom{30}})(x-3)\)
  4. \(\boxed{\phantom{30}}x^3 + 11x^2 - 17x + 6\) and \((\text-x+3)(2x-1)(x-2)\)
  5. \(6x^3+2x^2-16x+8\) and \((x-1)(2x+4)(\boxed{\phantom{30}}x-2)\)
  6. \(2x^3+7x^2-7x-12\) and \((2x-3)(x+\boxed{\phantom{30}})(x+\boxed{\phantom{30}})\)
  7. \(x^3+6x^2+\boxed{\phantom{30}}x-10\) and \((x+2)(x-1)(x+\boxed{\phantom{30}})\)

Student Response

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Activity Synthesis

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

Speaking, Representing: MLR8 Discussion Supports. Give students additional time to make sure that everyone in their group can explain how they found the missing numbers in the expressions and the relationship between standard and factored form. Invite students to rehearse with their group what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking. This will help students improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation

Lesson Synthesis

Lesson Synthesis

The purpose of this discussion is to compare a feature of polynomial long division that integer long division does not have, in addition to previewing some of the work in a later lesson on the meaning of the remainder when dividing polynomials.

Ask students to use long division to find the quotient for \(440 \div 24\). Invite students to share how they worked out the solution of 18 with a remainder of 8. Make sure students are clear on why they could not focus on just the leading digits, 2 and 4, to figure out what to put on top of the long division since 2 times 24 is larger than 44. Instead, you have to put a 1 on top in the tens place and then work out how many times 24 goes into 200. If students don’t remember the meaning of the remainder, write out that \(440 \div 24 = 18 \text{ R } 8\) is the same as saying \(440 = 24(18)+8\).

Next, ask students to use long division to find the quotient for \((4x^2+4x) \div (2x+4)\). Invite students to share how they worked out the solution of \(2x-2\) with a remainder of 8, focusing on how with polynomial long division you can just put the \(2x\) in the “\(x\)th” place, since \(2x \boldcdot 2x = 4x^2\), without worrying about the other terms. Conclude the discussion by asking students to write out a statement for the polynomial division similar to \(440 = 24(18)+8\) in order to make sense of what the remainder of 24 means (\(4x^2+4x = (2x+4)(2x-2)+8\)).

13.5: Cool-down - A Small Division Error (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

In earlier grades, we learned how to add, subtract, and multiply numbers. We also learned that one way to divide numbers, like 1573 divided by 11, is by using long division.

Long division.
Long division.
Long division.

Here the division has been completed in stages, focusing on the highest power of 10 (1,000) in the dividend 1,573, and working down. This long division shows that \(1573 = (11)(143)\).

Similar to integers, we can add, subtract, and multiply polynomials. It turns out that we can also use long division on polynomials. Instead of focusing on powers of 10, in polynomial long division we focus on powers of \(x\). Just as we started with the highest power or 10, we start with the highest power of \(x\), the leading term, and work down to the constant term. For example, here is \(x^3 + 5x^2 + 7x + 3\) divided by \(x+1\) completed in three stages. Notice how terms of the same degree are in the same columns.

\(\displaystyle \displaystyle \displaystyle \require{enclose} \begin{array}{r}   x^2\phantom{+7x+33} \\  x+1 \enclose{longdiv}{x^3+5x^2+7x+3} \\     \underline{\text-x^3-x^2} \phantom{+7x+333} \\   4x^2+7x \phantom{+33}\\   \end{array}\)

\(\displaystyle \displaystyle \displaystyle \require{enclose} \begin{array}{r}   x^2+4x\phantom{+33} \\  x+1 \enclose{longdiv}{x^3+5x^2+7x+3} \\     \underline{\text-x^3-x^2} \phantom{+7x+333} \\   4x^2+7x \phantom{+33}\\  \underline{\text-4x^2-4x} \phantom{+33} \end{array} \)

\(\displaystyle \displaystyle \displaystyle \require{enclose} \begin{array}{r}   x^2+4x+3 \\  x+1 \enclose{longdiv}{x^3+5x^2+7x+3} \\     \underline{\text-x^3-x^2} \phantom{+7x+333} \\   4x^2+7x \phantom{+33}\\  \underline{\text-4x^2-4x} \phantom{+33} \\ 3x+3 \end{array}\)

At each stage, the focus is only on the term with the largest exponent that’s left. At the conclusion, we can see that \(x^3 + 5x^2 + 7x + 3 = (x+1)(x^2+4x+3)\).