Download Review Synthetic Division to find roots of 3rd degree characteristic

Review SYNTHETIC DIVISION
to find roots of third degree
characteristic polynomial
Pamela Leutwyler
(2x – 5)(x + 3)(7x – 2) =
(2x – 5)(x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
The roots are:
5
2
-3
2
7
(2x – 5)(x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
The roots are:
5
2
-3
2
7
(2x – 5)(1x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
The roots are:
5
2
-3
2
7
(2x – 5)(1x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
The roots are:
5
2
-3
2
7
(2x – 5)(1x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
If
p
is a root of the polynomial equation
q
The roots are:
5
22
-3
1
2
77
(2x – 5)(1x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
If
p
is a root of the polynomial equation
q
Then q is a factor of 14
The roots are:
55
-3
-3
22
1
2
2
77
(2x – 5)(1x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
If
p
is a root of the polynomial equation
q
Then q is a factor of 14
and p is a factor of 30
A characteristic polynomial will always have lead coefficient = 1.
Rational eigenvalues will be integral factors of the constant coefficient
of the characteristic polynomial .
 1 3 3


example: find the eigenvalues for the matrix  2 2 3 
 4 2 1


3 
  1  3

 3
2
det  2   2  3   

4

 19
14  0




 4
 characteristic polynomial

2


1


potential rational roots are factors of 14.
+1, -1, +2, -2, +7, -7, +14, -14
3
2


4

 19
14  0




potential rational roots are factors of 14.
+1, -1, +2, -2, +7, -7, +14, -14
characteristic polynomial
Test the potrats using synthetic division:

1
-4
-19
-14
3
2


4

 19
14  0




potential rational roots are factors of 14.
+1, -1, +2, -2, +7, -7, +14, -14
characteristic polynomial
Test the potrats using synthetic division:
+1
1
1
-4
-19
-14
1
-3
-22
-3
-22
-36
The remainder is
NOT ZERO.
+1 is not a root.
3
2


4

 19
14  0




potential rational roots are factors of 14.
+1, -1, +2, -2, +7, -7, +14, -14
characteristic polynomial
Test the potrats using synthetic division:
+7
1
1
-4
-19
-14
7
21
14
3
2
0
The remainder is
ZERO.
+7 is a root.
3
2


4

 19
14  0




potential rational roots are factors of 14.
+1, -1, +2, -2, +7, -7, +14, -14
characteristic polynomial
Test the potrats using synthetic division:
+7
1
1
-4
-19
-14
7
21
14
3
2
0

 4

 19
14 


3
2
The remainder is
characteristic polynomial
ZERO.
+7 is a root.
2
factor this or use quadratic formula or continue with
(  7)(  3  2)synthetic division to get the other roots.