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82 Chapter 4 Complex Numbers Course Number Section 4.2 Complex Solutions of Equations Instructor Objective: In this lesson you learned how to determine the number of zeros of polynomial functions, and to find the zeros. I. The Number of Solutions of a Polynomial Equation (Pages 335−336) The Fundamental Theorem of Algebra implies that a polynomial equation of degree n has precisely n Date What you should learn How to determine the numbers of solutions of polynomial equations solutions in the complex number system. These solutions can be . . . real or complex and may be repeated. Example 1: How many zeros does the polynomial function f ( x) = 5 − 2 x 2 + x 3 − 12 x 5 have? 5 You can use a graph to check the number of real solutions of an equation. Every second-degree equation, ax 2 + bx + c = 0 , has precisely two solutions given by the Quadratic Formula. The expression b2 − 4ac is called the discriminant, and can be used to determine whether the solutions are real, repeated, or complex: 1) If the discriminant is less than zero, the equation has two complex solution(s). 2) If the discriminant is equal to zero, the equation has one repeated real solution(s). 3) If the discriminant is greater than zero, the equation has two distinct real solution(s). Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved. Section 4.2 83 Complex Solutions of Equations Example 2: Use the discriminant to find the number and type of solutions of the quadratic equation x 2 − 2x + 2 = 0 . Two complex solutions II. Finding Solutions of Polynomial Equations (Page 337) If the complex number a + bi (where b ≠ 0) is a solution of a What you should learn How to find the solutions of polynomial equations polynomial equation with real coefficients, then we know that a − bi is another solution of the equation. Example 3: Find the solutions of the quadratic equation x 2 − 2x + 2 = 0 . The solutions are 1 ± i. III. Finding Zeros of Polynomial Functions (Pages 338−339) The problem of finding the zeros of a polynomial function is essentially the same as . . . finding the solutions of a polynomial equation. The zeros of the polynomial function f ( x) = x 5 − 3 x 2 + 4 are simply . . . the solutions of the polynomial equation x5 − 3x2 + 4 = 0. Example 4: Find all the zeros of the polynomial function f ( x) = x 4 + 5 x 2 − 36 , given that 3i is a zero of f. Zeros: − 2, 2, − 3i, 3i Example 5: Find a fourth-degree polynomial function with real coefficients that has − 5, 5, and − 2i as zeros. f (x) = x4 − 21x2 − 100 Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved. What you should learn How to find the zeros of polynomial functions 84 Chapter 4 Complex Numbers Additional notes y y x y y x y x x y x x Homework Assignment Page(s) Exercises Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved.