1.  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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 4.4 Theorems about Zeros of Polynomial Functions

2. The Fundamental Theorem of Algebra Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Every polynomial function of degree n, with n  1, has at least one zero in the system of complex numbers.

3. The Fundamental Theorem of Algebra Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example: Find a polynomial function of degree 4 having zeros 1, 2, 4i, and  4i.

4. Zeros of Polynomial Functions with Real Coefficients Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Nonreal Zeros: If a complex number a + bi, b  0, is a zero of a polynomial function f(x) with real coefficients, then its conjugate, a  bi, is also a zero. (Nonreal zeros occur in conjugate pairs.) Irrational Zeros: If where a, b, and c are rational and b is not a perfect square, is a zero of a polynomial function f(x) with rational coefficients, then its conjugate is also a zero.

5. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Suppose that a polynomial function of degree 6 with rational coefficients has  3 + 2i,  6i, and as three of its zeros. Find the other zeros.

6. Rational Zeros Theorem Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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.

7. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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.

8. More Examples Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley