Download The Fundamental Theorem of Algebra

Complex Numbers
Lesson 3.3
The Imaginary Number i


1  i  i 2  1
By definition
Consider powers if i
i  1
2
i  i  i  i
3
2
i  i  i  1 1  1
4
2
2
i  i  i  1 i  i
5
...
4
It's any
number
you can
imagine
Using i

Now we can handle quantities that occasionally
show up in mathematical solutions
a  1  a  i a

What about
49
18
Complex Numbers

Combine real numbers with imaginary numbers

a + bi
Imaginary
part
Real part

Examples
3  4i
3
6  i
2
4.5  i  2 6
Try It Out

Write these complex numbers in standard form
a + bi
9  75
5  144
16  7
 100
Operations on Complex
Numbers

Complex numbers can be combined with





addition
subtraction
multiplication
division
Consider
9 12i    7 15i 
 3  i    8  2i 
 2  4i    4  3i 
Operations on Complex
Numbers

Division technique

Multiply numerator and denominator by the
conjugate of the denominator
3i
3i 5  2i


5  2i
5  2i 5  2i
15i  6i 2

25  4i 2
6  15i
6 15

  i
29
29 29
Complex Numbers on the
Calculator

Possible result

Reset mode
Complex format
to Rectangular

Now calculator does
desired result
Complex Numbers on the
Calculator

Operations with complex on calculator
Make sure to use the
correct character for i.
Use 2nd-i
Warning
16  49

Consider

It is tempting to combine them
16  49  16  49  4  7  28


The multiplicative property of radicals only works for
positive values under the radical sign
Instead use imaginary numbers
16  49  4i  7i  4  7  i 2  28
Try It Out

Use the correct principles to simplify the
following:
3  121
4 

81  4  81

3 
144

2
The
Discriminant

Consider the expression under the radical in the
quadratic formula
2


b  b  4ac
2a
This is known as the discriminant
What happens when it is




Positive and a perfect square?
Positive and not a perfect square?
Zero
Complex roots
Negative ?
Example

2
x
 3x  5  0
Consider the solution to

Note the graph


No intersections
with x-axis
Using the
solve and
csolve
functions
Fundamental Theorem of
Algebra

A polynomial f(x) of degree n ≥ 1 has at least
one complex zero


Number of Zeros theorem


Remember that complex includes reals
A polynomial of degree n has at most n distinct zeros
Explain how theorems apply to these graphs
Conjugate Zeroes Theorem

Given a polynomial with real coefficients
P( x)  an x  an 1 x
n

n 1
 ...  a1 x  a0
If a + bi is a zero, then a – bi will also be a
zero
Assignment



Lesson 3.3
Page 211
Exercises 1 – 78 EOO