Zeros of Polynomial FnsAlgebra III, Sec. 2.5

Objective

Determine the number of rational and real zeros of polynomial functions, and find the zeros.

Important Vocabulary

Upper Bound – no zeros are greater than this number

Lower Bound – no real zeros are less than this number

pg 162

The Fundamental Theorem of Alg

The Fundamental Theorem of Algebra guarantees that, in the complex number system, every nth-degree polynomial function has _______________ zeros.

precisely n

Example (on your handout)

How many zeros does the polynomial function have?

What’s the degree of the

function?How many zeros does the function

have? 5

The Fundamental Thm of Alg (cont)

The Linear Factorization Theorem states that… {formal definition on pg 154}The Linear Factorization Theorem tells us that an nth-degree polynomial can be factored into _______________ linear factors.

precisely n

Example 1Find all the zeros of each function. a.) How do you find the zeros of a function?Set =0 & solve…

Example 1Find all the zeros of each function. b.) Set =0 & solve…

Example 1Find all the zeros of each function. c.) Set =0 & solve…

Example 1Find all the zeros of each function. d.)

Set =0 & solve… difference of squares

difference of squares again

The Rational Zero Test Define purpose of the Rational Zero Test.

To help you find the possible rational zeros using the leading coefficient and the constant

State the Rational Zero Test Rational Zero = p = factor of the constant term q = factor of the leading coefficient

To use the Rational Zero Test… First, list all factors of the constant and then the leading coefficient. Then, set up p/q and test your possible zeros to find the actual zeros.

Example (on your handout)

List the possible rational zeros of the polynomial function.

pqp = ± 1, 5 q = ± 1, 3

Example 2Find the rational zeros of

Example 3Find the rational zeros of

Example 4Find the rational zeros of

Example 5Find all the real solutions of

Conjugate Pairs

Let f(x) be a polynomial function that has real coefficients. If a + bi is a zero of the function, then we know that _________________ is also a zero of the function.

a – bi

Example 6Find a third-degree polynomial function with integer coefficients that has 2, 7i, and -7i as zeros.

Factoring a Polynomial

To write a polynomial of degree n > 0 with real coefficients as a product without complex factors, write the polynomial as… The product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros.

Example 7Find all the zeros of

given that 1 + 4i is a zero of f.

Example 8Write

as the product of linear factors, and list all of its zeros.

Example (on your handout)

Write

as the product of linear factors, and list all of its zeros.

Explain why a graph cannot be used to locate complex zeros.