Review SYNTHETIC DIVISION to find roots of third

Review SYNTHETIC DIVISION to find roots of third

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 potential rational roots are factors of 14. +1, -1, +2, -2, +7, -7, +14, -14

3 4 2 19 14 0 characteristic polynomial Test the potrats using synthetic division: 1 -4 -19 -14 potential rational roots are factors of 14. +1, -1, +2, -2, +7, -7, +14, -14

3 4 2 19 14 0 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. potential rational roots are factors of 14. +1, -1, +2, -2, +7, -7, +14, -14 3 4 2 19 14 0 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. potential rational roots are factors of 14. +1, -1, +2, -2, +7, -7, +14, -14 3 4 2 19 14 0 characteristic polynomial Test the potrats using synthetic division: +7 1

1 3 -4 -19 -14 7 21 14 3 2 0 2

4 19 14 characteristic 2 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.

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