##
* *Reflection: Connection to Prior Knowledge
The Fundamental Theorem of Algebra - Section 2: The Fundamental Theorem of Algebra

A great way to help students understand something is by way of analogy. Essentially, it goes like this: "Just as A is to B, so is X to Y." If the students *really* understand how A relates to B, and if they *really* understand how X and Y are like A and B, then they can understand how X relates to B.

In this case, I'll build on the fact that my students know how to factor positive integers. Better yet, we'll talk about prime factorization: 30 = 6*5 = 2*3*5. Since 2, 3, and 5 are prime factors, the number 30 cannot be factored any further. More importantly, *no other number* can be factored as 2*3*5. Each integer has a unique prime factorization. This is the Fundamental Theorem of Arithmetic. Easy, right?

Well, the Fundamental Theorem of Algebra is very similar. If we think of a polynomial function as an integer, then we can think of linear binomials as prime numbers. Something like (x - 5) cannot be factored into two polynomials, so it's "prime". Just like every integer can be written as a unique product of prime numbers, so every polynomial can be written as a unique product of linear factors over the complex numbers.

Viola! Not only have we reviewed the Fundamental Theorem of Algebra, but we've approached it from a different angle, and thought of it in connection to something already very familiar. For some students, the feeling is the same as when you suddenly recognize a landmark and realize you aren't lost anymore!

*Teaching by Analogy*

*Connection to Prior Knowledge: Teaching by Analogy*

# The Fundamental Theorem of Algebra

Lesson 7 of 8

## Objective: SWBAT explain the Fundamental Theorem of Algebra as a natural consequence of the Factor Theorem. SWBAT identify both real & complex zeros of polynomials.

## Big Idea: Just how many solutions does this polynomial have?! Exactly the same number as its degree!

*50 minutes*

#### Finding Complex Zeros

*15 min*

Class will begin with a discussion of how to find the zeros of a cubic function that does *not* have three real roots. My notes for the details of this discussion are included as a resource. It's important to do this right because it's laying the foundation for **the Fundamental Theorem of Algebra** and bringing together a number of different concepts.

I'll begin with the equation **P(x) = x^3 + 4x^2 -2x - 20** on the board and ask the students how we should go about finding roots. I expect that many students will suggest looking at the graph, so we'll do that using GeoGebra or some other graphing software. The graph will clearly show that the function has only 1 real root at or near *x* = 2. We'll use synthetic substitution confirm that *x* = 2 is a zero of the function and then carefully show that there are exactly two more complex roots.

This class example is a good review of yesterday's lesson, but prepares students to move further. First, they will see (if they hadn't already) that the process of synthetic substitution not only confirms that (*x - b*) is a factor of the polynomial, but yields the coefficients of the other factor, as well. Second, they will have seen a good example of how the acceptance of complex roots allows us to identify exactly *n* zeros for an *n*-degree polynomial. This is exactly what we need to begin discussing the Fundamental Theorem of Algebra!

#### Resources

*expand content*

Now that students have seen (another) example of how complex roots/factors can be found, it's time to introduce the Fundamental Theorem of Algebra. By this point, most students should be able to see intuitively that something like the Fundamental Theorem must be true, but it's important to make it explicit for everyone.

In the resources, I've included a page of notes on the line of argument I like to take with my classes. I use Socratic questioning to develop a justification for the theorem based on the characteristics of odd and even polynomials. Along the way, it's important to punctuate the conversation with all sorts of quick examples to show, for instance, why a cubic function couldn't have more than three linear factors.

As a general rule, I avoid drawing any conclusions for my students. If they aren't seeing what follows from the statements we've made so far, I try to draw their attention to different pieces of the puzzle more explicitly and ask more and more pointed questions. Initially, I might ask, "We said before that every odd polynomial* must* have at least one real root. So, do odd polynomials have exactly *n* complex roots?"

If the students aren't able to answer, I'll have to be more specific. "If one root is real, is it possible to factor every odd polynomial in some way?" Assuming I get the correct response from the class, I'd then go on to ask, "So, one factor is linear; what must be true about the other factor? Is the other factor an odd or even polynomial? If it's an even polynomial, how many complex roots must it have?" This last question is very pointed and should lead every student to recall the previous step in which we showed that every even polynomial must have exactly *n* roots. Hopefully, I won't have to be this pointed because I want to leave as much of the thinking to the students as possible!

#### Resources

*expand content*

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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