## 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

Unit 4: Higher-Degree Polynomials
Lesson 7 of 8

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

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Subject(s):
Math, fundamental theorem of algebra, Polynomial Roots, factoring polynomial expressions, Algebra 2, master teacher project, prime factorization, Polynomial, Polynomial Operations and Functions, Socratic Dialogue
50 minutes

### Jacob Nazeck

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