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

3. (2x – 5)(x + 3)(7x – 2) =

4. 14x 3 + 3x 2 – 107x + 30 = 0

5. (2x – 5)(x + 3)(7x – 2) = 14x 3 + 3x 2 – 107x + 30 = 0 The roots are: -3

6. (2x – 5)(1x + 3)(7x – 2) = 14x 3 + 3x 2 – 107x + 30 = 0 The roots are: -3

7. (2x – 5)(1x + 3)(7x – 2) = 14x 3 + 3x 2 – 107x + 30 = 0 The roots are: -3

8. (2x – 5)(1x + 3)(7x – 2) = 14x 3 + 3x 2 – 107x + 30 = 0 The roots are: -3 If is a root of the polynomial equation

9. (2x – 5)(1x + 3)(7x – 2) = 14x 3 + 3x 2 – 107x + 30 = 0 The roots are: -3 If is a root of the polynomial equation Then q is a factor of 14 2 1 7

10. (2x – 5)(1x + 3)(7x – 2) = 14x 3 + 3x 2 – 107x + 30 = 0 The roots are: -3 If 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

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

12. 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

13. 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 -22 -36 The remainder is NOT ZERO. +1 is not a root.

14. 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.

15. 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. factor this or use quadratic formula or continue with synthetic division to get the other roots.