Section 2 5 The Fundamental Theorem of Algebra

Section 2. 5 The Fundamental Theorem of Algebra 1

Objective By following instructions students will be able to: § Use the Fundamental Theorem of Algebra to determine 2 the number of zeros. § Find all zeros of polynomial functions, including complex zeros. § Find conjugate pairs of complex zeros. § Find zeros of polynomial by factoring.

Definitions 3

Example 1: 4 Identify the number of zeros. a) The first degree polynomial . b) Counting multiplicity, the second degree polynomial function.

Example 2: Identify the number of zeros. a) The third degree polynomial function. b) The fourth degree polynomial function. 5

Example 3: Write As the product of linear factors, and list all of its zeros. 6

U-TRY #1 Find all the zeros of the function and write the polynomial as a product of linear factors. a) b) 7

Complex Zeros Occur in Conjugate Pairs 8

Example 4: Find a fourth-degree polynomial function, with real coefficients, that has as zeros. 9

Example 5: Write the polynomial a. As the product of factors that are irreducible over the rationals. b. As the product of linear factors and quadratic factors that are irreducible over the reals, and c. In completely factored form. 10

Example 6: Find all the zeros of Given that 1+3 i is a zero of f. 11

U-TRY #2 Find a polynomial function with integer coefficients that has the given zeros. (There are many correct answers) a) b) 12

Revisit Objective Did we… § Use the Fundamental Theorem of Algebra to determine 13 the number of zeros? § Find all zeros of polynomial functions, including complex zeros? § Find conjugate pairs of complex zeros? § Find zeros of polynomial by factoring?

Homework § HW: I: pg 134# 10 pg 180 #58 14 § II: pg 187#s 9 -12, 17, 19, 25, 27, 33 -49 ODD, 53 § III: pg 188 #s 68