1. M3U3D4 Warm Up Divide using Synthetic division: (2x ³ - 5x² + 3x + 7) /(x - 2) 2x² - x + 1 + 9/(x-2)

2. Homework Check: Document Camera

3. QUIZ

5. Complex Numbers

6. Definition of pure imaginary numbers: Any positive real number b, where i is the imaginary unit and bi is called the pure imaginary number.

7. Definition of pure imaginary numbers: i is not a variable it is a symbol for a specific number

8. Simplify each expression.

9. Remember Simplify each expression. Remember

10. Distribute Imaginary Numbers Handout

11. Simplify. To figure out where we are in the cycle divide the exponent by 4 and look at the remainder. 0

12. Simplify. Divide the exponent by 4 and look at the remainder. 1

13. Simplify. Divide the exponent by 4 and look at the remainder. 2

14. Simplify. Divide the exponent by 4 and look at the remainder. 3

15. Definition of Complex Numbers Any number in form a+bi, where a and b are real numbers and i is imaginary unit.

16. Definition of Equal Complex Numbers Two complex numbers are equal if their real parts are equal and their imaginary parts are equal. If a + bi = c + di, then a = c and b = d

17. When adding or subtracting complex numbers, combine like terms.

18. Simplify.

20. Multiplying complex numbers. To multiply complex numbers, you use the same procedure as multiplying polynomials.

21. Simplify. F O I L

22. F O I L

23. How to… Divide complex numbers Remember when we needed to divide radical expressions in Math 1?? How did we do that?!?! As a refresher, how would you divide the following: 4 2 + √3. 2 - √3 = 8 - 4√3 4 - 3 = 8 - 4√3 = 1 The conjugate of 2 + √3!

24. How to… Divide complex numbers Multiply the numerator & denominator by the conjugate! Then complete your steps for multiplying complex numbers! (a + bi) has a conjugate of (a – bi) and (a – bi) has a conjugate of (a + bi) Goal NO IMAGINARY NUMBERS IN THE DENOMINATOR!!

25. How to… Divide complex numbers i = √-1 i ² = -1 i ² CANNOT be in the simplified answer!

26. Example #1 Divide complex numbers Write the quotient in standard form. 7 + 5i 1  4i 7 + 5i 1 – 4i 7 + 5i 1 – 4i = 1 + 4i Multiply numerator and denominator by 1 + 4i, the complex conjugate of 1 – 4i. 7 + 28i + 5i + 20i 2 1 + 4i – 4i – 16i 2 = Multiply using FOIL. 7 + 33i + 20(–1) 1 – 16(–1) = Simplify and use i 2 = -1. –13 + 33i 17 = Simplify. 13 17 – =+ 33 17 i Write in standard form.

27. GUIDED PRACTICE 1. 1 + 9i i(9 – i) 2. (3 + i)(5 – i) 16 + 2i 3. 5 1 + i 5 2 – 5 2 i 11 13 + 16 13 i 4.5 + 2i 3 – 2i Write the expression as a complex number in standard form. ANSWER

28. Classwork U4D4 Complex Numbers

29. Homework U4D4 What do you call... odds And What do you get from… odds