Download 2.5 2.6Complex Numbers Bl 4.notebook

2.5 2.6Complex Numbers Bl 4.notebook
March 08, 2013
WARM UP
With your group
make a list of the
number systems that
you know about.
Mar 5­7:17 AM
Natural Numbers N
Integers Z
Real Number System R
Rational Numbers Q
Irrational numbers
Complex Number System C
Real Numbers
Imaginary Numbers
Feb 24­1:17 AM
Nov 1­10:21 AM
Find the solutions of
x2=10
10
The solutions are irrational roots
These are still real roots because the
number can be found on the number
line.
Oct 13­3:04 PM
Let's think about this
Real numbers contain the square roots of
positive numbers only.
To find roots of negative numbers we
define √-1 =i
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2.5 2.6Complex Numbers Bl 4.notebook
The complex number system is the set of all
real numbers and imaginary numbers. The
system consist of all expressions in the
form a + bi with the following properties
• a and b are real numbers
• i2 = -1
• You can use addition and multiplication as
if a + bi is a polynomial.
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To add or subtract the Complex numbers
you must first add the real parts and
then the imaginary parts
The additive identity: 0 + 0i
The additive inverse: -a - bi
a + bi + (-a - bi) = 0 + 0i = 0
Mar 6­7:09 AM
Multiply as you would two polynomials by using
distribution and combining like terms
Multiply
(3 + 5i) (2 - 3i)
Mar 6­7:10 AM
The complex cunjugate of the complex
number z = a + bi is z = a + bi = (a - bi)
What happens when you add a number with its conjugate?
What happens when you multiply a number with its conjugate?
Mar 6­7:10 AM
Oct 6­10:04 PM
Food for thought...
Dividing Complex Numbers
(21 + i)
(2 ­ i)
Multiply top and bottom by conjugate (21 + i ) ( 2 + i )
= (2 ­ i ) ( 2 + i )
(42 + 2i +21i + i2)
( 4 ­ i2)
=
(41 + 23i)
5
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Oct 12­8:41 AM
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2.5 2.6Complex Numbers Bl 4.notebook
March 08, 2013
Real numbers are two dimensional
Every Real number can be found on the
number line.
Complex Numbers are three dimensional
Every Complex Number can be found on the
complex plane
Real numbers are placed along the
horizontal axis and the imaginary number
along the vertical axis. The axis are call the
real axis and the imaginary axis
Mar 7­10:06 AM
A Great Test Question
Oct 6­10:01 PM
Plot the Points
imaginary axis
3 + 2i
­1 + 4i
2 ­ i
­4 ­3i
Oct 6­10:04 PM
Homework...
Precalc p 218­219 # 18­52 by 3’s
Mar 7­10:07 AM
real axis
Oct 6­10:05 PM
WARM UP...
Given f(x) = x2 - 3x evaluate the following:
f(3)
Solutions
0
f(-2)
10
-1 -3i
-3+i
f(i )
-5-15i
f(2 + i )
f(4 - 3i )
Nov 3­9:05 AM
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2.5 2.6Complex Numbers Bl 4.notebook
Goal:Complex Zeros
Fundamental Theorem Of Algebra
Fundamental Theorem Of Algebra
A Polynomial of degree n > 0 has n complex zeros. Some of the zeros may be repeated
Linear Factor Theorem
If f(x) is a polynomialof degree n>0 then f(x) has precisely n linear factors and
f(x) = a(x­z1)(x­z2)(x­z3)..........(x­zn)
where a is the leading coefficient of f(x) and z1,z2,z3, .....zn are the complex zeros of f(x). The zi are not necessarily distinct numbers; some may repeat`
Mar 3­1:25 PM
Find the cubic function that has
zeros of 5, 2i, and -2i
March 08, 2013
Look at the polynomial
f(x)= (x ­ 2i)(x+2i)
The quadratic function (x2+4) has two zeros:x=2i and x=­2i. Because the zeros are not real the graph has no x­intercepts.
If k is a nonreal complex number:
x=k is a solution (or root) of the equation f(x) = 0
k is a zero of the function f
(x­k) is a factor of f(x)
Note: Because it is not real they are not x­intercepts
Mar 3­1:35 PM
Important: If a + bi is a zero the a bi must also be a zero.
Do p 225 # 1, 5, 9, 11
#1 zeros: i, -i
5. zeros 2,3,i
Mar 3­1:57 PM
Mar 3­1:59 PM
Write is standard form
9. 1 (multiplicity 2) -2 (multiplicity of 3)
11. 2(multiplicty 2) -2 - i (multiplicity of 1
y = (x-2)2 (x + 2 + i) (x - 2 - i)
y=
Mar 4­1:07 PM
Mar 3­2:03 PM
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2.5 2.6Complex Numbers Bl 4.notebook
Do p 225 # 13-16
Use the graph #17, 19, 21
b,c,d,a
17 2 complex zeros, no real zeros
19. 3 complex zero, 1 real
21. 4 complex zeros, 2 real
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Polynomial Review
Homework :
p 225 # 23,25,30,33,37
Homework :
p 225 # 23,25,30,33,37
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2.5 2.6Complex Numbers Bl 4.notebook
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Mar 3­3:27 PM
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Mar 3­3:34 PM
Homework: p225 #
35,36,38,45, 46
Mar 3­4:26 PM
Mar 7­12:43 PM
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2.5 2.6Complex Numbers Bl 4.notebook
March 08, 2013
Goal: Solving
Inequalities with
polynomials
Mar 7­12:54 PM
To solve an inequality f(x) > 0 is to
find all the values of x that make f(x)
positive.
To solve an inequality f(x) < 0 is to
find all the values of x that make f(x)
negative.
Mar 3­4:30 PM
Find where the polynomial is zero, positive
and negative.
Let f(x) = (x-2)(x2+1)(x-3)2
f(x) = 0 at x = 2, x=3
(-)(+)(+)
(+)(+)(-)
2
f(x) < 0 (-∞,2)
(+)(+)(+)
3
f(x) > 0 (3, ∞)
f(x) = 0 x=2,1
Mar 4­9:20 AM
Mar 4­9:21 AM
This is consistent with what we have
learned about the multiplicity of the
factor. When the multiplicity is odd,
look for a sign change. When the
multiplicity is even - no change in sign
Do p 246 # 1,3,5, 6
Mar 4­9:22 AM
Mar 7­1:09 PM
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2.5 2.6Complex Numbers Bl 4.notebook
March 08, 2013
Class work p246 #7-12
Homework p246 #13,15,18,20
Concept worksheet
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