The purpose of this lesson is for students to understand the Remainder Theorem. A focus of the lesson is that for any polynomial \(p(x)\) and linear expression \((xa)\), we can state that \(p(x) = (xa)q(x) + r\), where \(r\) is the remainder and \(q(x)\) is a polynomial. To get to this endpoint, the start of the lesson asks students to consider the meaning of the remainder when dividing with integers, explicitly making the connection between dividend, divisor, quotient, and remainder written using long division and written as a multiplication equation.
In the following activity, students identify an unknown coefficient in a polynomial that has a known factor. Students use a consequence of the Remainder Theorem studied previously: if a polynomial \(p(x)\) has a factor of \((xa)\), then \(p(a)=0\). Using new connections made during their work with division and writing multiplication equations, students now consider this fact from the perspective of what must be true about \(p(x)=(xa)q(x)+r\) when we know \(r=0\). This activity is purposely unscaffolded to encourage students to make sense of the problem and choose a solution path (MP1).
In the last activity, students complete repeated calculations to help them make the connection that the value of \(p(a)\) is equal to the value of the remainder when \(p(x)\) is divided by \((xa)\) (MP8). Once the Remainder Theorem is established, it can then be stated that, for a polynomial \(p(x)\), \(p(a)=0\) means \((xa)\) must be a factor. In previous lessons, students used zeros to predict factors, and now they will know that zeros always correspond to factors in this way.
Lesson overview
 15.1 Warmup: Notice and Wonder: Division Leftovers (5 minutes)

15.2 Activity: The Unknown Coefficient (10 minutes)
 Includes "Are you Ready for More?" extension problem
 15.3 Activity: A Study of Remainders (20 minutes)
 Lesson Synthesis
 15.4 Cooldown: Using Remainder Knowledge (5 minutes)
Learning goals:
 Comprehend that for a polynomial \(p(x)\) and a number \(a\), the remainder on division by \((xa)\) is \(p(a)\).
 Comprehend that for a polynomial \(p(x)\), \((xa)\) is a factor if \(p(a)=0\) and, conversely, that \(p(a)=0\) if \((xa)\) is a factor.
Learning goals (student facing):
 Let’s learn about the Remainder Theorem.
Learning targets (student facing):
 I understand the remainder theorem and why it's true.
Standards:
 This lesson builds on the standard: CCSS.HSAAPR.A.1MS.AAPR.1MO.A1.APR.A.1
 This lesson builds towards the standard: CCSS.HSAAPR.B.2MS.AAPR.2MO.A2.APR.A.2
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