Reflection: Grappling with Complexity The Fundamental Theorem of Algebra and Imaginary Solutions - Section 4: Structured Discussion and Notes on the Fundamental Theorem

 

If students are paying attention, they should take issue with a ‘fundamental’ theorem that states a truth in direct opposition to what they have learned in the past. For one, it’s been drilled into them that the discriminant in the quadratic formula can identify whether the function has one root, two roots, or no real roots. In addition, they will have seen many graphical counterexamples that will challenge their visual understanding of what ‘roots’ of a polynomial look like.

Wrapping their minds around a rule that says that the number of roots is equal to the degree can be a very complex idea at this point in their studies. I find that it can help to tread carefully, acknowledge (and even congratulate) confused looks, and then remind them that all of mathematics is designed to bring order to chaos. It’s a much more elegant idea that a fourth degree polynomial will have four roots - more elegant than saying it might have four or three or two or one or none! The introduction of a new definition, i = sqrt(-1), makes this idea a truth (and not just for fourth degree polynomials)! Keeping this in perspective can help motivate the learning of our new ‘imaginary’ piece of content.

  It Doesn't Seem So Fundamental!
  Grappling with Complexity: It Doesn't Seem So Fundamental!
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The Fundamental Theorem of Algebra and Imaginary Solutions

Unit 4: Polynomial Theorems and Graphs
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.

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