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

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

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

Example 1 Find 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. p q p = ± 1, 5 q = ± 1, 3

Example 2 Find the rational zeros of

Example 3 Find the rational zeros of

Example 4 Find the rational zeros of

Example 5 Find 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 6 Find 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 7 Find all the zeros of given that 1 + 4i is a zero of f.

Example 8 Write 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.