1. 2.5 Descartes’ Rule of Signs To apply theorems about the zeros of polynomial functions To approximate zeros of polynomial functions

2. Descartes’ Rule of Signs The number of positive real zeros of a polynomial function P(x), with real coefficients, is equal to the number of variations in sign of the terms of P(x) or is less than this number by a multiple of 2. The number of negative real zeros is equal to the number of variations in sign of the terms of P(-x) or is less than this number by a multiple of 2.

3. Example 1 Find the possible number of positive and negative real zeros of The problem isn’t asking for the zeros themselves, but what are the possible number of them. This can help narrow down the possibilities when you do go on to find the zeros.

4. Example 2 Find the possible number of positive and negative real zeros of

5. Upper and Lower Bounds Let P(x), a polynomial function with positive leading coefficient, be divided by x – c. If c > 0 and all coefficients in the quotient and remainder are nonnegative, then c is an upper bound of the zeros. If c < 0 and the coefficients in the quotient and remainder alternate in sign, then c is a lower bound of the zeros.

6. Example 3 Show that all real roots of the equation Lie between –4 and 4

7. The Intermediate Value Theorem If f(x) is a polynomial function and f(a) and f(b) have different signs, then there is at least one value, c, between a and b such that f(c)=0.

8. This works because 0 separates the positives from the negatives. So to go from positive to negative or vice a versa you would have to hit a point in between that goes through 0.

9. Example 4 Show that has real zeros between 3 and 4 and 0 and -1.

10. Example 5 Find the upper and lower bounds. Find the possible number of positive and negative zeros. Find the location of the zeros.

11. Example 6 Find the upper and lower bounds. Find the possible number of positive and negative zeros. Find the location of the zeros.

12. Summary Descartes’ Rule of Signs is used to determine the number of possible positive and negative zeros. The Upper and Lower Bound Theorem is used to determine the bounds of the zeros. The Intermediate Value Theorem is used to determine between which real numbers a zero occurs.