1. Chapter 2 Polynomial and Rational Functions

2. Warm Up 2.3  An object is launched at 19.6 meters per second from a 58.8 -meter tall platform. The equation for the object’s height s at time t seconds after launch is where s is in meters. When does the object strike the ground? 2

3. 2.3 Real Zeros of Polynomial Functions Objectives:  Use long division to divide polynomials by other polynomials.  Use synthetic division to divide polynomials by binomials of the form ( x – k ).  Use the Remainder and Factor Theorems.  Use the Rational Zero Test to determine possible rational zeros of polynomial functions.  Use Descartes’s Rule of Signs and the Upper and Lower Bound Rules to find zeros of polynomials. 3

4. Consider This ….  Consider the graph of the function One zero is x = ___________. The linear factor is ______________.  Use the linear factor to produce a complete factorization of the function. That is, factor the function completely. Then find the zeros. 4

5. Polynomial Long Division  Divide x 2 +3x +5 by x + 1. Identify the dividend, divisor, quotient, and remainder. 5

6. Notes on Division  Before performing division, be sure to: 1. Write the dividend and divisor in descending powers of the variable, that is, in standard polynomial form. 2. Insert zeros for any missing powers of the variable. 6

7. Examples  Use polynomial long division to solve. 7

8. Synthetic Division  A shortcut for polynomial long division.  Divisor must linear, that is, x – k.  Example: Divide x 4 – 10x 2 – 2x +4 by x + 3 8

9. Examples  Use synthetic division to solve. 9

10. The Remainder Theorem  If a polynomial f (x) is divided by x – k, the remainder is r = f (k).  To evaluate a polynomial function at a value x = k, divide the polynomial by x – k. The remainder will equal f (k). 10

11. Examples  Evaluate the function at x = 2.  Evaluate the function at x = –4. 11

12. The Factor Theorem  A polynomial f (x) has a factor (x – k) if and only if f (k) = 0.  We can demonstrate that some (x – k) is a factor of f (x) by dividing and showing that the remainder is equal to zero. 12

13. Example  Show that (x – 2) and (x + 3) are factors of Then find the remaining factors of the function. 13

14. Real Zeros  The real zeros of a polynomial can be rational or irrational.  If a polynomial has integer coefficients, we can find the potential rational zeros by using the factors of the leading coefficient a n and the constant term a 0. 14

15. The Rational Zero Test 15

16. Using the Rational Zero Test  Find a 0 and a n.  List all factors (positive and negative) of a 0 and a n.  Write the possible rational zeros as a quotient:  Check to see which of these numbers are zeros by using synthetic division.  Continue factoring. 16

17. Helpful Hint  Graph the function on your calculator FIRST to eliminate unlikely possible zeros. 17

18. Example  Find the rational zeros of  What is the constant term?  What is the leading coefficient?  What are the factors of each?  Write the possible rational zeros.  Use synthetic division to find the first zero.  Factor completely. 18

19. Example  Find the rational zeros of  What is the constant term?  What is the leading coefficient?  What are the factors of each?  Write the possible rational zeros.  Since the list is so long, use calculator to narrow down the search.  Use synthetic division to find the first zero.  Factor completely. 19

20. Homework 2.2  Worksheet 2.2 20