Chapter 3: Polynomial Functions

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. 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 • bis called the imaginary part

4. 3.1 Examples of Complex Numbers Write (a) (b)

5. 3.1 Products and Quotients Involving Negative Radicands • CAUTION Rewrite as before using any other rules for radicals. • 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. 3.1 Operations with Complex Numbers • All properties of real numbers are extended to complex numbers. Example Adding and Subtracting Complex Numbers • (b) Analytic Solution

7. 3.1 Multiplying Complex Numbers Example Multiply Analytic Solution Graphing Calculator Solution Figure 6, pg. 3-5

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. 3.1 Complex Conjugates The conjugate of the complex number is Their product is • The table shows several pairs of conjugates and their products.

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