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The Complex Number System Name: ____TEACHER COPY ______ _______ CCSS.Math.Content.HSN.CN.C.9 Date: ______________________ Period: _____ Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Lesson Plan: 1. Hook: Radical Sign: I love you! So why can't we be together? Complex Number: It's complex. 2. Introduction and Vocabulary: In order to understand and simplify complex numbers, we need to first discuss some vocabulary used with them: real numbers, coefficients, radical sign, square root, commutative, associative, distributive, binomial, trinomial, factor, polynomial, imaginary numbers and complex numbers. A wonderful introductory lecture on the fundamental theorem of algebra can be found at https://www.khanacademy.org/math/algebra2/polynomial_and_rational/fundamental-theorem-ofalgebra/v/fundamental-theorem-of-algebra-intro 3. Guided Practice / Review: Factor the following expressions using real or complex factors. 1. x2 + 9 2. x2 + 4x + 3 3. x2 + 36 4. 4x2 - 3x + 2 Check student work and review any discrepancies. 4. Independent Practice : Students will review Fundamental Theorem of Algebra and the quadratic formula and apply them in problems that involve complex number solutions. After reviewing as a group, have the students work independently on the specific examples. Check student work and review any discrepancies. 5. Exit Slip: Find the real and complex roots of the function p(x) = 4x2 - 3x + 2. Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. All Rights Reserved by NewMathTeacher.Net The Complex Number System Name: ____________________ ______ _______ CCSS.Math.Content.HSN.CN.C.9 Date: ______________________ Period: _____ Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Fundamental Theorem of Algebra The Fundamental Theorem of Algebra states that any polynomial of degree n has n roots. These roots (or solutions) may be real solutions or complex solutions. 𝑝(𝑥) = 𝑎𝑥 𝑛 + 𝑏𝑥 𝑛−1 + ⋯ + 𝑘 Let's focus on using this theorem with quadratics (n = 2). Quadratic Formula For an equation in the form: Use the discriminant as a quick way to tell how many and what type of solutions you will have: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 Use the values of a, b, and c to solve for x: 𝑥= −𝑏 ± √𝑏 2 Discriminant − 4𝑎𝑐 2𝑎 Example: The discriminant (D) is: D = b2 – 4ac If D > 0, then there are 2 real solutions If D = 0, then there is 1 real solutions If D < 0, then there are 2 complex solutions Use the discriminant to test how many solutions the equation to the left has: 3x2 + 5x - 2 = 0 D = b2 – 4ac = a= b= c= Solve each equation. Include complex solutions. x2 + 32 = 0 2. 6x2 + 24 = 0 3. x2 + 36 = 0 4. x2 + 100 = 0 5. 3x2 + 108 = 0 6. x2 + 12 = 0 7. x2 - 3x + 3 = 0 8. x2 + 5x + 4 = 0 9. 4x2 - 3x + 2 = 0 1. Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. All Rights Reserved by NewMathTeacher.Net