1. Bell Ringer

2. Multiplying and Dividing with Complex Numbers Monday, February 29, 2016

3. Multiplying Complex Numbers  When we multiply complex numbers, sometimes the imaginary part “disappears”.  Ex: (2 + 3i) (2 – 3i) =  (4 – 6i + 6i – 9i 2 ) =  (4 + 9) = 15

4. Theorems say…  Complex roots come in pairs…  Ex: You can’t have a root of 3i without a root of -3i

5. Multiplying Complex Numbers  If a polynomial has complex roots, when you multiply the factors to find the original polynomial, the imaginary part “disappears”.

6. What about dividing complex numbers?  You can’t divide by an imaginary number.  To make the denominator real, we must multiply by a special 1 called the CONJUGATE.  We make the middle term “disappear”.  6 – 3i ÷ 2 + 4i  6 – 3i (2 – 4i) 2 + 4i (2 – 4i)  12 – 30i + 12i 2 4 – 16i 2  12 – 30i + (-12) 4 – (-16)  - 30i 20  -3 i 2

7. Complex numbers and Conjugates  Complex numbers are in the form a + bi  Complex conjugates are in the form a - bi  Complex: 2 + 4i  Conjugate: 2 – 4i  Complex: -3 + 5i  Conjugate: -3 – 5i  Complex: 1 – 7i  Conjugate: 1 + 7i

8. Time to Practice!  Classwork: Find Polynomial Functions given the roots (multiplying)  Homework: Dividing Complex Numbers (multiplying by i and complex conjugates)

9. Exit Ticket  1. What happens when you multiply complex conjugates?  2. Both complex and irrational roots must come in _____.  3. How do you make the denominator real if it is a complex number?