## Proofs of the Remainder Theorem.pdf - Section 3: Justifying the Conjecture

# The Remainder Theorem, Day 2 of 2

Lesson 4 of 8

## Objective: SWBAT use the remainder theorem to identify roots of polynomial equations. SWBAT explain the relationship between factors and zeros of a polynomial equation.

## Big Idea: By observing regular patterns in their work, students formulate conjectures leading to the Remainder Theorem for polynomials.

*45 minutes*

Having completed problems 1 - 3 of the handout, *The Remainder Theorem*, students are ready to formulate conjectures that will lead to a formal statement of the theorem.

A proof of the theorem is developed through Socratic questioning in a whole-class discussion, and then students use the theorem to answer a variety of questions about an unknown polynomial.

#### Resources

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#### Formulating a Conjecture

*15 min*

During the previous lesson (or at home) students should have completed problems 1 - 3 on *The Remainder Theorem*. Now, it's time to begin formulating a conjecture about remainders and function values based on the results of problem #3. This is problem #4 on the handout.

Using a Think-Pair-Share strategy, I will give the class about 2 minutes to formulate a conjecture on their own, then another 2 minutes to work with a partner to refine their conjecture, and then I will call on several students to share their conjecture. For strategies for building a class discussion around student input, please see my **Strategy Folder**.

My first concern is that students identify the correct relationship - *the remainder is equal to the value of the function* - and my second is that they express it carefully. The class will compare & debate various forms of the conjecture, I will encourage them to say precisely what they mean (**MP 6**), and together we will agree on a single wording. Of course, I'll be quietly guiding them toward something very close to the formal expression of the Remainder Theorem that they will see in their textbook.

In the end, this conjecture will be written clearly on the board for all to students to copy into their notes.

#### Resources

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#### Justifying the Conjecture

*15 min*

Together, the class will do its best to explore the truth of the conjecture. (**MP 3**) I will open the conversation with something like this: "Wow. This is really strange! Why on earth would the remainder be the same as the function value? I mean, we have four examples in which this is clearly true, but how can we claim that it's *always* true?"

Many students will recall from previous courses (and from common sense) that simply piling on more examples doesn't make a proof. What we need is some kind of general justification, some kind of explanation of *why* this phenomenon occurs. Depending on the class, this conversation can become very sophisticated. To begin, it might be worth comparing the steps in the long division process with the steps in the evaluation process. Next, you might consider taking a generic quadratic equation and dividing it by a generic binomial (see the resource). Finally, you could walk your students through a more general proof like the second one included in the resource.

Somewhere along the way, be sure to point out (or guide some student to point it out for you) that they've already proven a weaker version of this conjecture: the Factor Theorem, in which the remainder & function value were both zero.

Once the conjecture has been adequately justified, I'll erase the word "Conjecture" from the board and replace it with "The Remainder Theorem." Everyone should copy the theorem into their notes.

#### Resources

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#### Using the Remainder Theorem

*15 min*

Students will work on problem #5 individually for 5 minutes, then in small groups for the remainder (Get it?) of the lesson. The objective is to complete problems 1 through 5 and to be prepared to turn in the assignment at the beginning of class tomorrow. Please see the video resource for more details!

An exit ticket might be an appropriate formative assessment for the end of the lesson.

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: Polynomials & Place Value
- LESSON 2: Polynomial Long Division
- LESSON 3: The Remainder Theorem, Day 1 of 2
- LESSON 4: The Remainder Theorem, Day 2 of 2
- LESSON 5: Higher Degree Polynomials, Day 1 of 2
- LESSON 6: Higher Degree Polynomials, Day 2 of 2
- LESSON 7: The Fundamental Theorem of Algebra
- LESSON 8: Polynomial Practice & Review