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# The Fundamental Theorem of Algebra and Imaginary Solutions

Lesson 5 of 15

## Objective: SWBAT divide polynomial expressions and solve polynomial equations; SWBAT understand that polynomial equations sometimes have imaginary solutions.

## Big Idea: The degree of a polynomial equation tells us how many solutions to expect as long as we include both real and imaginary solutions.

*90 minutes*

In Quiz: Factoring Solving, Remainder Theorem, students have the opportunity to demonstrate mastery of factoring higher order polynomials, using the Remainder Theorem to identify roots of a polynomial function, and using the Zero Product Property to solve polynomial equations [MP1].

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#### Post-Quiz Work

*15 min*

While my students complete their quiz, I write a message on the board about what they should do when they are finished. I want them to take out the homework that was due the previous day, WS Solving Polynomials with the Remainder Theorem. In a different color from the one they used to complete the assignment (hopefully pencil!) I tell them that for each of the four problems, they should write (a) the degree of the polynomial expression in the equation and (b) the number of solutions. They should then consider if there is a pattern to their answers [MP8].

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After students have completed the quiz and had a chance to examine their homework for a pattern in the degree and number of solutions, I ask students to explain any patterns they may have found. It is likely that they will see that the number of solutions is equal to the degree because the equations they were given have all unique, real solutions. We discuss this pattern informally and I then put the following equation on the board for students to solve: x^{2}+4=0. From this, students will see that the number of real solutions is not equal to the degree of the equation, which seems to run counter to the pattern we just discovered.

I write out the version of the **Fundamental Theorem of Algebra** commonly presented in Algebra 2 textbooks, which is that the degree of a polynomial equation is equal to the number of complex solutions, provided that repeated solutions are counted separately. I underline "complex" and the last phrase and write in some explanation of these. I explain that the set of complex numbers includes all the numbers they have learned about and some more. I explain that repeated solutions come from two of the same factors and provide an example with a repeated solution like (x-3)(x-3)=0 [MP6].

I then provide explicit notes on * i *and simplifying radicals with a negative radicand. I do not yet spend time simplifying powers of

**or performing operations on complex numbers because this will take some time.**

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#### Exit Ticket and Homework

*15 min*

As an exit ticket, I ask my students to close their notebook and write the Fundamental Theorem of Algebra in their own words on an index card. For homework, they will solve some polynomial equations with imaginary solutions in Solving Equations with Complex Solutions [MP1]. I make solutions to this worksheet available to my students on Edmodo.

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- LESSON 1: Seeing Structure in Expressions - Factoring Higher Order Polynomials
- LESSON 2: Proving Polynomial Identities
- LESSON 3: Polynomial Long Division and Solving Polynomial Equations
- LESSON 4: The Remainder Theorem
- LESSON 5: The Fundamental Theorem of Algebra and Imaginary Solutions
- LESSON 6: Arithmetic with Complex Numbers
- LESSON 7: Review of Polynomial Roots and Complex Numbers
- LESSON 8: Quiz and Intro to Graphs of Polynomials
- LESSON 9: Graphing Polynomials - End Behavior
- LESSON 10: Graphing Polynomials - Roots and the Fundamental Theorem of Algebra
- LESSON 11: Analyzing Polynomial Functions
- LESSON 12: Quiz on Graphing Polynomials and Intro to Modeling with Polynomials
- LESSON 13: Performance Task - Representing Polynomials
- LESSON 14: Review of Polynomial Theorems and Graphs
- LESSON 15: Unit Assessment: Polynomial Theorems and Graphs