## Notes Polynomial Theorems and Imaginary Numbers.docx - Section 3: Explanation: Complex Zeros

*Notes Polynomial Theorems and Imaginary Numbers.docx*

# Polynomials with Complexes… Complex Zeros that is!

Lesson 10 of 15

## Objective: SWBAT identify the imaginary zeros of a polynomial function and use imaginary roots to find the standard form of a polynomial function.

## Big Idea: Personal response systems keep students engaged and monitor their current progress as they find all zeros of a polynomial function.

*50 minutes*

#### Warm-up: clicker questions

*10 min*

At the start of class, students should complete the warm-up questions on Flipchart - complex zeros and more practice (p.1-3). This question on page 2 was intended for students to apply the remainder theorem. However, I would definitely encourage students to solve the problem using any method that makes sense to them. Similarly, the question on page 3 was intended for students to apply the factor theorem. After all students have submitted their answers, I want to take a minute or two to talk with students about the different methods they used/we could have used to solve these problems and what they would have looked like: synthetic division, long division, dividing using squares, or the most efficient methods, applying the remainder or factor theorem. The following questions would be used to help lead the discussion:

*How would we set up these problems?**Where would we find the remainder?**If it is a factor what would the remainder be?**How are roots, zeros, and factors related?**How could we use our calculator as a tool for efficiency?*

Here is another great opportunity to bring our students attention to the mathematical practices… **Mathematical Practice 5: Use appropriate tools strategically** - Depending on how this unit plays out (and their responses to the last question), I may take some time for calculator tips here. I just want to be sure that my students understand how they could find the zeros of a polynomial function on the calculator. Specifically by using the Calculate: Zeros option on the graph menu. I will require students to still show their work of synthetic division or long division, but I think it is important they have a method to check their answers too.

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Take some time today to address as many questions from the 3-2-1 assessments as you’d like. I am planning on answering a bulk of the questions today and maybe a few more over the next few class periods. I think it is important to answer or at least address all of the questions in some way or another. Sometimes I may just answer some of the questions where students just seem really lost with a “Whoever wrote this please see me in advisory so we can talk further about this.” The students who are just so utterly lost in the unit will definitely show through on these 3-2-1 assessments because they won’t even be able to state what they have a question on. We probably already know who these kids are from all of the built in checks for understanding throughout the unit, but just in case we missed them this is a great time to intervene and try to motivate the student to come in for help before the upcoming tests and projects. This may be a good time to mention to students that today is the last ‘teacher led’ day. Students will be doing all of the work for the next 5 class periods. We, the teachers, have been working too hard and talking way too much the last few days and now it’s time for students to apply all that knowledge!

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**Preparation:** For the closure part of today’s lesson students will need individual whiteboards and markers.

**Narrative:** On the Flipchart - complex zeros and more practice (p.4-9) there are a variety of questions to assess students’ current progress. I plan to have students complete their work on a whiteboard and then just text in their answer to let me know they are done. Note that the problem on pages 4-5 is the same, so students should use the fact that (x+5) is a factor to find all the zeros of the function on page 5. Again, it is important to continue to emphasize how factors, zeros, x-intercepts, and roots are all related. I chose to cover up the multiple choice on the last two questions because I feel that the multiple choice will make it too easy. I do plan to show the choices eventually. But I want students to first work it out the long way without being able to test the zeros and use the process of elimination. If you have more time in your class period, you can also have students complete more practice problems from yesterday’s extra practice worksheet. I will link that here again.

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: Puzzling Polynomials and Quizzical Quadratics
- LESSON 2: Building Connections: Building Polynomials (Day 1 of 2)
- LESSON 3: Building Connections: Building Polynomials (Day 2 of 2)
- LESSON 4: Detailed Descriptions of our Puzzling Polynomials
- LESSON 5: Got zeros? Polynomials do! Multiplicity of Zeros (Day 1 of 3)
- LESSON 6: Got zeros? Polynomials do! Multiplicity of Zeros (Day 2 of 3)
- LESSON 7: Got zeros? Polynomials do! Multiplicity of Zeros (Day 3 of 3)
- LESSON 8: Long, Synthetic, and Square! Oh my! Polynomial Division
- LESSON 9: Rational Roots and Remainders: Important Theorems of Polynomials
- LESSON 10: Polynomials with Complexes… Complex Zeros that is!
- LESSON 11: Putting the Pieces of Polynomials Together (Day 1 of 2)
- LESSON 12: Putting the Pieces of Polynomials Together (Day 2 of 2)
- LESSON 13: Roller Coaster Polynomials
- LESSON 14: Test Review
- LESSON 15: Polynomials Unit Test