# 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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