Lesson 13
Polynomial Division (Part 2)
13.1: Notice and Wonder: Different Divisions (10 minutes)
Warmup
The purpose of this warmup 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 wholeclass discussion.
Student Facing
What do you notice? What do you wonder?
\(\displaystyle \displaystyle \displaystyle \require{enclose} \begin{array}{r} 2x^2\phantom{+7x+22} \\ x+1 \enclose{longdiv}{2x^3+7x^2+7x+2} \\ \underline{\text2x^32x^2} \phantom{+7x+22} \\ 5x^2+7x \phantom{+22} \end{array} \)
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:
\(\displaystyle \displaystyle \displaystyle \require{enclose} \begin{array}{r} 2x^2+5x+2 \\ x+1 \enclose{longdiv}{2x^3+7x^2+7x+2} \\ \underline{\text2x^32x^2} \phantom{+7x+22} \\ 5x^2+7x \phantom{+22} \\ \underline{\text5x^25x} \phantom{+22} \\ 2x+2 \\ \underline{\text2x2} \\ 0 \end{array} \)
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 nonzero 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.
Supports accessibility for: Language; Socialemotional skills
Student Facing
 Diego used the long division shown here to figure out that \(6x^27x5 = (2x+1)(3x5)\). Show what it would look like if he had used a diagram.
\(\displaystyle \displaystyle \require{enclose} \begin{array}{r} 3x5 \\ 2x+1 \enclose{longdiv}{6x^2  7x  5} \\ \underline{\text6x^2  3x} \phantom{55} \\ \text10x5 \\ \underline{10x+5} \\ 0 \end{array} \)
\(2x\) \(6x^2\) 1 Pause here for a wholeclass discussion.

\((x2)\) is a factor of \(2x^3  7x^2 + x + 10\), which means there is some other factor \(A\) where \(2x^3  7x^2 + x + 10=(x2)(A)\). Finish the division started here to find the value of \(A\).
\(\displaystyle \displaystyle \require{enclose} \begin{array}{r} 2x^2 \phantom{+x+100} \\ x2 \enclose{longdiv}{2x^37x^2+x+10} \\ \underline{\text2x^3+4x^2} \phantom{+x+100} \end{array}\)
 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?
 What is \((x^41) \div (x1)\)?
 Use your response to predict what \((x^71) \div (x1)\) 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.
Design Principle(s): Optimize output (for explanation); Maximize metaawareness
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 wholeclass 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.
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.

\(A(x)=x^3  7x^2  16x + 112\), \((x7)\)
\(\displaystyle \displaystyle \require{enclose} \begin{array}{r} x^2\phantom{55555555555} \\ x7 \enclose{longdiv}{x^3  7x^2 16x+112} \\ \underline{\textx^3 + 7x^2} \phantom{.55555555555} \\ \end{array} \)  \(C(x)=x^3  3x^2  13x + 15\), \((x+3)\)
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.
Design Principle(s): Maximize metaawareness
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 standardform 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 wholeclass discussion.
Student Facing
Here are pairs of equivalent expressions, one in standard form and the other in factored form. Find the missing numbers.
 \(x^2 + 9x + 14\) and \((x+2)(x+\boxed{\phantom{30}})\)
 \(x^2  9x + 20\) and \((x\boxed{\phantom{30}})(x\boxed{\phantom{30}})\)
 \(2x^2+2x24\) and \(2(x+\boxed{\phantom{30}})(x3)\)
 \(\boxed{\phantom{30}}x^3 + 11x^2  17x + 6\) and \((\textx+3)(2x1)(x2)\)
 \(6x^3+2x^216x+8\) and \((x1)(2x+4)(\boxed{\phantom{30}}x2)\)
 \(2x^3+7x^27x12\) and \((2x3)(x+\boxed{\phantom{30}})(x+\boxed{\phantom{30}})\)
 \(x^3+6x^2+\boxed{\phantom{30}}x10\) and \((x+2)(x1)(x+\boxed{\phantom{30}})\)
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.
Design Principle(s): Support sensemaking; 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 \(2x2\) 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 (\(4x^2+4x = (2x+4)(2x2)+8\)).
13.5: Cooldown  A Small Division Error (5 minutes)
CoolDown
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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.
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.
\(\displaystyle \displaystyle \displaystyle \require{enclose} \begin{array}{r} x^2\phantom{+7x+33} \\ x+1 \enclose{longdiv}{x^3+5x^2+7x+3} \\ \underline{\textx^3x^2} \phantom{+7x+333} \\ 4x^2+7x \phantom{+33}\\ \end{array}\)
\(\displaystyle \displaystyle \displaystyle \require{enclose} \begin{array}{r} x^2+4x\phantom{+33} \\ x+1 \enclose{longdiv}{x^3+5x^2+7x+3} \\ \underline{\textx^3x^2} \phantom{+7x+333} \\ 4x^2+7x \phantom{+33}\\ \underline{\text4x^24x} \phantom{+33} \end{array} \)
\(\displaystyle \displaystyle \displaystyle \require{enclose} \begin{array}{r} x^2+4x+3 \\ x+1 \enclose{longdiv}{x^3+5x^2+7x+3} \\ \underline{\textx^3x^2} \phantom{+7x+333} \\ 4x^2+7x \phantom{+33}\\ \underline{\text4x^24x} \phantom{+33} \\ 3x+3 \end{array}\)
At each stage, the focus is only on the term with the largest exponent that’s left. At the conclusion, we can see that \(x^3 + 5x^2 + 7x + 3 = (x+1)(x^2+4x+3)\).