1. Chapter 2 Polynomial and Rational Functions

2. Warm Up 2.4  From 1980 to 2002, the number of quarterly periodicals P published in the U.S. can be modeled by  Where t is the number of years since 1980. a. Describe the end behavior of the graph. b. Graph the model on the domain 0 ≤ t ≤ 22. c. Use the model to predict the number of quarterly periodicals in the year 2010. 2

3. 2.4 Complex Numbers Objectives:  Use the imaginary unit i to write complex numbers.  Add, subtract, and multiply complex numbers.  Use complex conjugates to write the quotient of two complex numbers in standard form.  Plot complex numbers in the complex plane. 3

4. The Imaginary Unit, i  If we find the square root of a number, are there any restrictions?  Yes – the term inside the radical must be ≥ 0.  What happens if the term inside the radical is negative?  For example, what if x 2 + 1 = 0 ?  This equation has no real solution.  The Imaginary Unit i was created for this purpose.  That is, and 4

5. Complex Numbers  A complex number consists of a real part and an imaginary part.  Standard form of a complex number is a ± bi, where a is the real part and bi is the imaginary part. 5

6. Adding and Subtracting Complex Numbers  Add or subtract the real parts and add or subtract the imaginary parts. 6

7. Examples  Perform the following operations. Simplify if possible. 7

8. Multiplying Complex Numbers  FOIL or distribute and simplify.  Note: Be sure numbers are written in complex form before you begin. 8

9. Examples  Perform the following operations. Simplify if possible. 9

10. Imaginary Units of Higher Power  How can we easily find i raised to any integer power?  What is the next one?  What is the pattern? 10

11. Complex Conjugates  The product of two complex numbers is usually another complex number.  However, the product of two complex numbers can be a real number if the two numbers are conjugates.  Numbers of the form a + bi and a – bi are complex conjugates. 11

12. Example  Multiply 3 – 5i by its complex conjugate. 12

13. Complex Number Quotients  When we divide a complex number by another complex number, we can write it as a quotient.  We want to make sure that the denominator is real, not imaginary or complex.  To do so, we must multiply both the numerator and the denominator of the quotient by the complex conjugate of the denominator. 13

14. Example  Write the quotient in standard form. 14

15. Homework 2.4  Worksheet 2.4 #5, 9, 11, 17 – 35 odd, 37, 45 – 61 odd 15