review synthetic division to find roots of third degree characteristic polynomial pamela leutwyler
TRANSCRIPT
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
(2x – 5)(x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
The roots are:
2
5
7
2-3
(2x – 5)(1x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
The roots are:
2
5
7
2-3
(2x – 5)(1x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
The roots are:
2
5
7
2-3
(2x – 5)(1x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
The roots are:
2
5
7
2-3
q
pIf is a root of the polynomial equation
(2x – 5)(1x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
The roots are:
2
5
7
2-3
q
pIf is a root of the polynomial equation
Then q is a factor of 14
2 1 7
(2x – 5)(1x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
The roots are:
2
5
7
2-3
q
pIf is a root of the polynomial equation
Then q is a factor of 14 and p is a factor of 30
2 1 7
5-3
2
A characteristic polynomial will always have lead coefficient = 1.
Rational eigenvalues will be integral factors of the constant coefficient of the characteristic polynomial .
example: find the eigenvalues for the matrix
124
322
331
014194
124
322
331
det 23
polynomialsticcharacteri
potential rational roots are factors of 14. +1, -1, +2, -2, +7, -7, +14, -14
014194 23 polynomialsticcharacteri
potential rational roots are factors of 14. +1, -1, +2, -2, +7, -7, +14, -14
Test the potrats using synthetic division:
1 -4 -19 -14
014194 23 polynomialsticcharacteri
potential rational roots are factors of 14. +1, -1, +2, -2, +7, -7, +14, -14
Test the potrats using synthetic division:
+1 1 -4 -19 -14
1
1
-3
-3
-22
-22
-36
The remainder is NOT ZERO.+1 is not a root.
014194 23 polynomialsticcharacteri
potential rational roots are factors of 14. +1, -1, +2, -2, +7, -7, +14, -14
Test the potrats using synthetic division:
+7 1 -4 -19 -14
1
7
3
21
2
14
0
The remainder is ZERO.+7 is a root.
014194 23 polynomialsticcharacteri
potential rational roots are factors of 14. +1, -1, +2, -2, +7, -7, +14, -14
Test the potrats using synthetic division:
+7 1 -4 -19 -14
1
7
3
21
2
14
0
)23)(7(
14194
2
23
polynomialsticcharacteri
The remainder is ZERO.+7 is a root. factor this or use quadratic formula or continue with
synthetic division to get the other roots.