1. The Original f(x)=x 3 -9x 2 +6x+16 State the leading coefficient and the last coefficient Record all factors of both coefficients According to the Fundamental Theorem of Calculus, how many complex zeros does this function have? Use graphing calculators to find the zeros of function. Would you want to use synthetic division to find these zeros? Why or Why not?

3. 0=x 2 +10x+16

4. f(x)=x 3 -9x 2 +6x+16 Zeros: x=-1,2,8

5. Integral Root Theorem Let x n +a 1 x n-1 +…+a n - 1 x+a n =0 represent a polynomial equation that has a leading coefficient of 1, integral coefficients, and a n ≠0. Any rational roots of this equation must be integral factors of a n.

6. f(x)=6x 2 -5x-4

7. Rational Root Theorem Let a 0 x n +a 1 x n-1 +…+a n-1 +a n =0 represent a polynomial equation of degree n with integral coefficients. If a rational number p/q where p and q have no common factors, is a root of the equation, then p is a factor of a n and q is a factor of a 0

8. f(x)=6x 2 -5x-4 Zeros: x=-1/2, 4/3

9. Possible Rational Roots f(x)=7x 2 +4x-1 y=-3x 5 -16x 4 +15 f(t)=2t 4 -2t 6 +4t 2 -t+10 g(x)=7x 2 +6x 3 -x

10. Descartes’ Rule of Signs Given the polynomial P(x) written in descending powers of x. The number of positive real zeros of P(x) is the same as the number of changes in sign of the coefficients of the terms (or less than this by an even number). The number of negative real zeros of P(x) is the same as the number of changes in sign of the coefficients of the terms of P(-x) (or less than this by an even number)

11. Descartes’ Rule of Signs Simplified… If you put a polynomial in descending order of power, then the number of positive real roots is the number of changes in the signs of the coefficients of f(x) (or 2 less, or 4 less, or …) The number of negative real roots is the number of changes in the signs of the coefficients of f(-x) (or 2 less, or 4 less, or …)

12. Descartes’ Rule with “The Original” f(x)= x 3 -9x 2 +6x+16 Zeros: x=-1,2,8

13. Find the number of positive and negative real zeros of f(x)=2x 5 +3x 4 -6x 3 +6x 2 -8x+3

14. f(x)=2x 5 +3x 4 -6x 3 +6x 2 -8x+3 Graph of function

15. In your journal… Given equation f(x)= 2x 5 + 3x 4 - 6x 3 + 6x 2 - 8x + 3 Find the possible zeros If you know x = -3 is a zero, what other negative zeros might there be? Find the remaining real zeros. How do you know you have found all of the real zeros?