2. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial Division; The Remainder and Factor Theorems 4.4 Theorems about Zeros of Polynomial Functions 4.5 Rational Functions 4.6 Polynomial and Rational Inequalities

3. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 4.4 Theorems about Zeros of Polynomial Functions  Find a polynomial with specified zeros.  For a polynomial function with integer coefficients, find the rational zeros and the other zeros, if possible.  Use Descartes’ rule of signs to find information about the number of real zeros of a polynomial function with real coefficients.

4. Slide 4.4 - 4 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Rational Zeros Theorem Let where all the coefficients are integers. Consider a rational number denoted by p/q, where p and q are relatively prime (having no common factor besides  1 and 1). If p/q is a zero of P(x), then p is a factor of a 0 and q is a factor of a n.

5. Slide 4.4 - 5 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Given f(x) = 2x 3  3x 2  11x + 6: a) Find the rational zeros and then the other zeros. b) Factor f(x) into linear factors. Solution: a) Because the degree of f(x) is 3, there are at most 3 distinct zeros. The possibilities for p/q are:

6. Slide 4.4 - 6 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example continued Use synthetic division to help determine the zeros. It is easier to consider the integers before the fractions. We try 1:We try  1: Since f(1) =  6, 1 is Since f(  1) = 12,  1 is not a zero.not a zero. –6–12–12 –12–12 6–11–321 12–6–52 65–2 6–11–32–1

7. Slide 4.4 - 7 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example continued We try 3:. We can further factor 2x 2 + 3x  2 as (2x  1)(x + 2). 0–232 –696 6–11–323 Since f(3) = 0, 3 is a zero. Thus x  3 is a factor. Using the results of the division above, we can express f(x) as

8. Slide 4.4 - 8 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example continued The rational zeros are  2, 3 and The complete factorization of f(x) is: