review synthetic division to find roots of third degree characteristic polynomial pamela leutwyler
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![Page 1: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/1.jpg)
Review SYNTHETIC DIVISION
to find roots of third degree characteristic polynomial
Pamela Leutwyler
![Page 2: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/2.jpg)
(2x – 5)(x + 3)(7x – 2) =
![Page 3: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/3.jpg)
(2x – 5)(x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
![Page 4: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/4.jpg)
(2x – 5)(x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
The roots are:
2
5
7
2-3
![Page 5: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/5.jpg)
(2x – 5)(1x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
The roots are:
2
5
7
2-3
![Page 6: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/6.jpg)
(2x – 5)(1x + 3)(7x – 2) =
14x3 + 3x2 – 107x + 30 = 0
The roots are:
2
5
7
2-3
![Page 7: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/7.jpg)
(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
![Page 8: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/8.jpg)
(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
![Page 9: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/9.jpg)
(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
![Page 10: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/10.jpg)
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
![Page 11: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/11.jpg)
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
![Page 12: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/12.jpg)
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.
![Page 13: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/13.jpg)
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.
![Page 14: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler](https://reader035.vdocuments.net/reader035/viewer/2022072015/56649eb75503460f94bc0946/html5/thumbnails/14.jpg)
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.