Transcript The Fundamental Theorem of Algebra

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