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
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The Fundamental Theorem of Algebra

Unit 4: Higher-Degree Polynomials
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!

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5 teachers like this lesson
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
cubic graph
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