Transcript Document
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
2
3
4
19
14 0
characteristic
potential rational roots are factors of 14.
+1, -1, +2, -2, +7, -7, +14, -14
polynomial
Test the potrats using synthetic division:
1
-4
-19
-14
2
3
4
19
14 0
characteristic
potential rational roots are factors of 14.
+1, -1, +2, -2, +7, -7, +14, -14
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.
2
3
4
19
14 0
characteristic
potential rational roots are factors of 14.
+1, -1, +2, -2, +7, -7, +14, -14
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.
2
3
4
19
14 0
characteristic
potential rational roots are factors of 14.
+1, -1, +2, -2, +7, -7, +14, -14
polynomial
Test the potrats using synthetic division:
+7
1
1
-4
-19
-14
7
21
14
3
2
0
4
19
14
3
2
characteristic
polynomial
The remainder is
ZERO.
+7 is a root.
factor this or use quadratic formula or continue with
( 7)( 3 2)synthetic division to get the other roots.
2