1. Descartes’ Rule of Signs
1. The number of possible positive real zeros is equal to the number of changes in the sign of the coefficients of f (x) or less by an even integer.
2. The number of possible negative real zeros is equal to the number of changes in the sign of the coefficients of f (-x) or less by an even integer.
Ex 1: Find the possible number of
positive and negative real zeros of
f (x) = 3x3 – 5x2 + 6x – 4.

2. Ex 2: Find the possible number of
positive and negative real zeros of
f (x) = x3 – 7x2 – 2x – 8.

3. Upper and Lower Bound rules
Let f (x) be a polynomial with real coefficients and a positive leading coefficient. Suppose f (x) is divided by x – c using synthetic division.
1. If c > 0 and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f.
2. If c < 0 and the numbers in the last row are alternately positive and negative (zero entries count as either positive or negative), c is an lower bound for the real zeros of f.

4. Ex 3: Use synthetic division to show that -2 is a lower bound of f and that 1 is an upper bound of f for f (x) = 5x3 – 3x + 12.
Ex 4: Use the strategies that we’ve learned to find the zeros of f (x) = 6x3 – 4x2 + 3x – 2.

5. Practice Assignment:
Section 2.3C (pg ) #