Transcript 3.4 and 3.5 Complex Numbers and Zeros

1.
You throw away the outside
and cook the inside.
Then you eat the outside
and throw away the inside.
What did you eat?:
corn
ANSWERS
Check Homework
PRE
Lesson 3.4 & 3.5 Complex Number
& Complex Zeros
Objectives:
•To understand complex numbers.
•To add, subtract & multiply complex
numbers.
•To solve equations with complex
numbers as solutions.
To reduce a power
of i, just divide by 4
and the remainder
will correspond to
one of these.
i  1
2
i  1
3
i  i
4
i  1
i  1
13
Special care must be taken when performing
calculations involving square roots of negative
numbers. Although
when
a and b are positive. This is not true when
both are negative. For example:
1
9 x  54  0
54 54
2
9 x  54
2
9
9
x  6
x   6
2
x   6  1
x  i 6
2a. P( x)  x  x  x
x(x 2  x  1)
x 0
x2  x 1 0
3


x 0

2
1 i 3
x
2
2b. P( x)  x3  7 x 2  18 x  18; x  3

3 1 7 18 18

3 12 18
1


4
6
0
2
(x

3)
(x
 4 x  6)

  
x  3 x  2  i 2
   

2c. P( x)  x  x  2 x  2; x  1, 1
4
2
1 1
0 1
1 1
1 1 0
2 2
0 2
2 0
(x  1)(x 3  x 2  2)

  
1 1 1 0 2
   1
 2 2


1 2 2
0

(x  1) (x  1)(x 2  2x  2)
  
 1 
x  1
x 
 i







Whenever a+bi is a zero, its
complex conjugate a-bi is also a zero:
3a. Degree 3 and zeros 2 and 3-i.
(x  2)(x  (3  i))(x  (3  i))



(x  2)(x  (3  i))(x  (3  i))


 


x 2
2 2

x 2 0
(x  2)


x  3i
(3 i) (3 i)
 x  (3  i)  0
x  (3  i)



3a. Degree 3 and zeros 2 and 3-i.
(x  2)(x  (3  i))(x  (3  i))
(x  2)(x  3 i)(x  3 i)





(x  2)(x 2  6x 10)

x 3  8x 2  22x  20
Classwork: Worksheet: 3.4 & 3.5