College Algebra Chapter 3 Polynomial and Rational Functions Section 3.4 Zeros of Polynomials

Concepts 1. Apply the Rational Zero Theorem 2. Apply the Fundamental Theorem of Algebra 3. Apply Descartes’ Rule of Signs 4. Find Upper and Lower Bounds

Apply the Rational Zero Theorem

Apply the Rational Zero Theorem

Example 1: List all possible rational zeros of

Example 2: Find the zeros and their multiplicities.

Example 2 continued:

Example 3: Find the zeros and their multiplicities.

Concepts 1. Apply the Rational Zero Theorem 2. Apply the Fundamental Theorem of Algebra 3. Apply Descartes’ Rule of Signs 4. Find Upper and Lower Bounds

Apply the Fundamental Theorem of Algebra If f (x) is a polynomial of degree n ≥ 1 with complex coefficients, then f (x) has at least one complex zero.

Apply the Fundamental Theorem of Algebra Linear Factorization Theorem:

Apply the Fundamental Theorem of Algebra Conjugate Zeros Theorem: If f (x) is a polynomial with real coefficients and if a + bi (b ≠ 0) is a zero of f (x) then its conjugate a – bi is also a zero of f(x).

Apply the Fundamental Theorem of Algebra Number of Zeros of a Polynomial: If f (x) is a polynomial of degree n ≥ 1 with complex coefficients, then f (x) has exactly n complex zeros provided that each zero is counted by its multiplicity.

Example 4: Given that is a zero of find the remaining zeros and factor as a product of linear factors.

Example 4 continued:

Example 5: Given that is one solution to , find the remaining solutions.

Example 5 continued:

Example 6: Find a polynomial f (x) of lowest degree with zeros of 3i and 2 (multiplicity 2).

Example 7: Find a polynomial p(x) of degree 3 with zeros – 2i and 4. The polynomial must also satisfy the condition that p(0) = 32.

Concepts 1. Apply the Rational Zero Theorem 2. Apply the Fundamental Theorem of Algebra 3. Apply Descartes’ Rule of Signs 4. Find Upper and Lower Bounds

Apply Descartes’ Rule of Signs Let f (x) be a polynomial with real coefficients and a nonzero constant term. Then, The number of positive real zeros is either the same as the number of sign changes in f (x) or less than the number of sign changes in f (x) by a positive even integer. The number of negative real zeros is either the same as the number of sign changes in f (–x) or less than the number of sign changes in f (–x) by a positive even integer.

Example 8: Determine the number of possible positive and negative real zeros. Number of possible positive real zeros Number of possible negative real zeros Number of imaginary zeros Total (including multiplicities)  

Concepts 1. Apply the Rational Zero Theorem 2. Apply the Fundamental Theorem of Algebra 3. Apply Descartes’ Rule of Signs 4. Find Upper and Lower Bounds

Find Upper and Lower Bounds A real number b is called an upper bound of the real zeros of a polynomial if all real zeros are less than or equal to b. A real number a is called a lower bound of the real zeros of a polynomial if all real zeros are greater than or equal to a.

Find Upper and Lower Bounds Let f (x) be a polynomial of degree n ≥ 1 with real coefficients and a positive leading coefficient. Further suppose that f (x) is divided by (x – c). 1. If c > 0 and if both the remainder and the coefficients of the quotient are nonnegative, then c is an upper bound for the real zeros of f (x). 2. If c < 0 and the coefficients of the quotient and the remainder alternate in sign (with 0 being considered either positive or negative as needed), then c is a lower bound for the real zeros of f.

Example 9: Given , show that 4 is an upper bound and –2 is a lower bound for the real zeros.