1. Copyright © 2007 Pearson Education, Inc. Slide 3-1

2. Copyright © 2007 Pearson Education, Inc. Slide 3-2 Chapter 3: Polynomial Functions 3.1 Complex Numbers 3.2 Quadratic Functions and Graphs 3.3 Quadratic Equations and Inequalities 3.4 Further Applications of Quadratic Functions and Models 3.5 Higher Degree Polynomial Functions and Graphs 3.6 Topics in the Theory of Polynomial Functions (I) 3.7 Topics in the Theory of Polynomial Functions (II) 3.8 Polynomial Equations and Inequalities; Further Applications and Models

3. Copyright © 2007 Pearson Education, Inc. Slide 3-3 3.1 Complex Numbers The complex number system is an extended number system that includes the real number system as a subset. Define Numbers of the form where a and b are real numbers are called complex numbers in standard form. –a is called the real part –b is called the imaginary part

4. Copyright © 2007 Pearson Education, Inc. Slide 3-4 Write (a) (b) 3.1 Examples of Complex Numbers

5. Copyright © 2007 Pearson Education, Inc. Slide 3-5 3.1 Products and Quotients Involving Negative Radicands CAUTION Rewrite as before using any other rules for radicals. (1) (2) Technology Note: Some graphing calculators such as the TI-83 are capable of complex number operations by setting the mode to a+bi. Verify (1) and (2).

6. Copyright © 2007 Pearson Education, Inc. Slide 3-6 3.1 Operations with Complex Numbers All properties of real numbers are extended to complex numbers. Example Adding and Subtracting Complex Numbers (a) (b) Analytic Solution (a) (b)

7. Copyright © 2007 Pearson Education, Inc. Slide 3-7 3.1 Multiplying Complex Numbers Example Multiply Analytic Solution Graphing Calculator Solution Figure 6, pg. 3-5

8. Copyright © 2007 Pearson Education, Inc. Slide 3-8 3.1 Powers of i Observe the following pattern. Any larger power of i is found by writing the power as a product of two powers of i, one exponent a multiple of 4. Example Simplify Solution

9. Copyright © 2007 Pearson Education, Inc. Slide 3-9 3.1 Complex Conjugates The table shows several pairs of conjugates and their products. The conjugate of the complex number is A complex number multiplied by its complex conjugate results in a real number.

10. Copyright © 2007 Pearson Education, Inc. Slide 3-10 3.1 Dividing Complex Numbers Procedure: Multiply the numerator and denominator by the complex conjugate of the denominator. Example Find the quotient and write in standard form. Analytic Solution Graphical Solution Figure 8 pg 3-7