Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler.

14
Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler

Transcript of Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler.

Page 1: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler.

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.

(2x – 5)(x + 3)(7x – 2) =

Page 3: Review SYNTHETIC DIVISION to find roots of third degree characteristic polynomial Pamela Leutwyler.

(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.

(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.

(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.

(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.

(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.

(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.

(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.

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.

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.

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.

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.

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.