The Fundamental Theorem of Algebra and Imaginary Solutions
Lesson 5 of 15
Objective: SWBAT divide polynomial expressions and solve polynomial equations; SWBAT understand that polynomial equations sometimes have imaginary solutions.
In Quiz: Factoring Solving, Remainder Theorem, students have the opportunity to demonstrate mastery of factoring higher order polynomials, using the Remainder Theorem to identify roots of a polynomial function, and using the Zero Product Property to solve polynomial equations [MP1].
While my students complete their quiz, I write a message on the board about what they should do when they are finished. I want them to take out the homework that was due the previous day, WS Solving Polynomials with the Remainder Theorem. In a different color from the one they used to complete the assignment (hopefully pencil!) I tell them that for each of the four problems, they should write (a) the degree of the polynomial expression in the equation and (b) the number of solutions. They should then consider if there is a pattern to their answers [MP8].
After students have completed the quiz and had a chance to examine their homework for a pattern in the degree and number of solutions, I ask students to explain any patterns they may have found. It is likely that they will see that the number of solutions is equal to the degree because the equations they were given have all unique, real solutions. We discuss this pattern informally and I then put the following equation on the board for students to solve: x2+4=0. From this, students will see that the number of real solutions is not equal to the degree of the equation, which seems to run counter to the pattern we just discovered.
I write out the version of the Fundamental Theorem of Algebra commonly presented in Algebra 2 textbooks, which is that the degree of a polynomial equation is equal to the number of complex solutions, provided that repeated solutions are counted separately. I underline "complex" and the last phrase and write in some explanation of these. I explain that the set of complex numbers includes all the numbers they have learned about and some more. I explain that repeated solutions come from two of the same factors and provide an example with a repeated solution like (x-3)(x-3)=0 [MP6].
I then provide explicit notes on i and simplifying radicals with a negative radicand. I do not yet spend time simplifying powers of i or performing operations on complex numbers because this will take some time.
Exit Ticket and Homework
As an exit ticket, I ask my students to close their notebook and write the Fundamental Theorem of Algebra in their own words on an index card. For homework, they will solve some polynomial equations with imaginary solutions in Solving Equations with Complex Solutions [MP1]. I make solutions to this worksheet available to my students on Edmodo.