= 0:
¸ = cos
¹
4
+ ¸ sin
¹
4
=
1
2
+
¸
2
.
= 1:
¸ = cos
¹
4
+ ¹
+ ¸ sin
¹
4
+ ¹
= −
1
2
−
¸
2
.
Third
quadrant.
Cos< 0,
sin
<0.
<>±
C
L
= :;<
C
L
= A/
4
So the square roots of
¸
are
±
1
2
+
¸
2
8/12/2015
7
ANOTHER EXAMPLE
Example 2:
Find all the cube roots of
8.
Solution:
Since
8 =
2
7
,
we know one already:
2
is a root
Let’s find the others.
Polar form of
8:
’ = 8,
) = 0.
8 = 8.
cos · =
8
8
= 1,
sin · =
0
8
= 0.
So
· = 0.
So
³ = 8 = 2
7
,
· = 0.
8
6
=
³
/
7
cos
·
3
+
2 ¹
3
+ ¸ sin
·
3
+
2 ¹
3
=
8
/
7
cos
0
3
+
2 ¹
3
+ ¸ sin
0
3
+
2 ¹
3
= 2
cos
2 ¹
3
+ ¸ sin
2 ¹
3
,
= 0,1,2.
= 0:
8
6
= 2[cos 0 + ¸ sin 0] = 2
1 + 0. ¸
= 2.
8/12/2015
8
8
6
=
2
cos
2 ¹
3
+ ¸ sin
2 ¹
3
,
= 0,1,2.
= 1:
8
6
= 2
cos
2¹
3
+ ¸ sin
2¹
3
= 2
cos
¹ −
¹
3
+ ¸ sin
¹ −
¹
3
= 2
−
1
2
+ ¸
3
2
= −1 + ¸
3.
= 2:
8
6
= 2
cos
4¹
3
+ ¸ sin
4¹
3
= 2
cos
¹ +
¹
3
+ ¸ sin
¹ +
¹
3
= 2
−
1
2
− ¸
3
2
= −1 − ¸
3
Second quadrant.
Cos< 0,
sin
>0.
<>±
C
5
=
5
4
, :;<
C
5
=
A
4
Third
quadrant.
Cos< 0,
sin
<0.
FACTORING
³
P
− *
We calculated
:
8
6
= 2,
−1 ± ¸
3.
Now if
Q
¶
= 0,
then
³ − ¶
is
a factor of
Q
³ .
(Remainder Theorem)
Now
³
7
− 8 = 0
when
³ = 2,
−1 ±
3.
So
³ − 2 ,
³ + 1 −
3 ,
and
³ + 1 +
3
Are all the factors of
³
7
− 8.
So
³
7
− 8 =
³ − 2
³ + 1 −
3
³ + 1 +
3 .
8/12/2015
9
STEPS IN FACTORING
³
P
− *
:
Step 1:
Write the number
*
in polar form:
* = ´[cos µ +
± sin µ].
Step 2
: Find the n values of
*:
·
*
·
=
*
/
P
$%&
µ
¸
+
2 °
¸
+ ± &±¸
µ
¸
+
2 °
¸
,
= 0,1, … , ¸ − 1.
Step 3:
If the ¶ values of
*
·
are
³
/
, ³

, … ³
P
,
Then
³
P
− * =
³ − ³
/
³ − ³

…
³ − ³
P
.
NOW YOU TRY IT:
Find the three cube roots of
−8.
Factorise
³
7
+ 8
into three factors.
8/12/2015
10
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 Complex number, Imaginary unit