standard_description: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
problem_activity: 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
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.
\(\displaystyle  \displaystyle \require{enclose} \begin{array}{r}   3x-5 \\   2x+1 \enclose{longdiv}{6x^2 - 7x - 5} \\      \underline{\text-6x^2 - 3x} \phantom{-55} \\    \text-10x-5 \\  \underline{10x+5} \\    0  \end{array} \)
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(2x\\)"}]}], "td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(6x^2\\)"}]}, {}]}, {"th": [{"_value": "1"}], "td": [{}, {}]}]}]}]}
Pause here for a whole-class discussion.
\((x-2)\)
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
\(A\)
.
\(\displaystyle \displaystyle \require{enclose} \begin{array}{r}   2x^2 \phantom{+x+100} \\   x-2 \enclose{longdiv}{2x^3-7x^2+x+10} \\      \underline{\text-2x^3+4x^2} \phantom{+x+100} \end{array}\)
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.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(x^2\\)"}]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(6x\\)"}]}, {"_value": "5"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(2x\\)"}]}], "td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(2x^3\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(12x^2\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(10x\\)"}]}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "1"}], "td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(x^2\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(6x\\)"}]}, {"_value": "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
\((x^4-1) \div (x-1)\)
?
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


aligns: no