1. Sec 3.4 & Sec 3.5 Complex Numbers & Complex Zeros
Objectives:
To understand complex numbers.
To add, subtract & multiply complex numbers.
To solve equations with complex numbers as solutions.

2. Complex Numbers
If , then A complex number is a number of the form where a and b are real numbers. The real part of the complex number is a and the imaginary part is bi.

3. Ex 1. State the real and imaginary parts of the following complex numbers.-2

4. Operations on Complex Numbers
To add complex numbers, add the real parts and add the imaginary parts. (a + bi) + (c + di) = (a + c) + (b + d)I To subtract complex numbers, subtract the real parts and subtract the imaginary parts. (a + bi) – (c + di) = (a – c) + (b – d)i

5. Multiply complex numbers like binomials, using
(a + bi) . (c + di) = (ac – bd) + (ad + bc)i

6. Ex 2. Perform the indicated operation and write the result as a + bi.
(3 + 5i) + (4 – 2i)
(3 + 5i) – (4 – 2i)
(3 + 5i)(4 – 2i)

7. Powers of Any power of i can be reduced to one of the following:
To reduce a power of i, just divide by 4 and the remainder will correspond to one of these. For example,
Because 23/4 has a remainder of 3 so the answer corresponds to

8. Ex 3. Evaluate.
a) b) c) d)

9. Class Work
Perform the indicated operation and write your answer in a + bi form.
(2 + 5i) + (4 – 6i)
(5 – 3i)(1 + i)

10. Square Roots of Negative Numbers
Just as every positive real number r has two square roots , every negative number has two square roots as well.
If -r is a negative number, then its square roots are , because:
and

11. If –r is negative, then the principal square root of –r is: For example:

12. Ex 4. Evaluate the expression.

13. Ex 5. Evaluate and write the result in the form of a + bi.

14. Class Work
Evaluate the expression and write your result in the form of a + bi

15. Complex Roots of Quadratic Equations
We know that if b2 – 4ac < 0 then the quadratic has no real solutions. However, in the complex number system, the equation will always have solutions. The solutions will be complex numbers and will have the form a + bi and a – bi. The solutions will always come in pairs, called conjugates.

16. Ex 6. Solve each equation.
x2 + 9 = 0
x2 + 4x + 5 = 0

17. Class Work
Find all the zeros of the polynomial

18. Ex 7. Find a polynomial with integer coefficients that have the given zeros.
Degree 3 and zeros 2 and i.
Degree 2 and zeros 1 + i, and 1 – i.

19. HW Worksheet 3.4 & 3.5