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
Loading resource...
 

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!

  Print Lesson
4 teachers like this lesson
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
cubic graph
 
1
2
3
Similar Lessons
 
Rational Functions and Inequalities Formative Assessment
12th Grade Math » Polynomial and Rational Functions
Big Idea: Use a Quick Quiz to see if students understand rational functions and inequalities.
  Favorites(0)
  Resources(6)
Troy, MI
Environment: Suburban
Tim  Marley
 
Choosing a Method to Find x-intercepts
Algebra I » Quadratics!
Big Idea: Students take a step back from their work to examine a variety of quadratic functions and reflect on why they might choose one method over another.
  Favorites(3)
  Resources(12)
Boston, MA
Environment: Urban
Amanda Hathaway
 
Building Connections: Building Polynomials (Day 2 of 2)
12th Grade Math » Polynomial Functions and Equations
Big Idea: This amazing NCTM lesson continues as students now work backwards to find the linear factors of polynomial graphs and extend it to third degree polys.
  Favorites(1)
  Resources(20)
Phoenix, AZ
Environment: Urban
Tiffany Dawdy
 
Something went wrong. See details for more info
Nothing to upload
details
close