2.5 Zeros of Polynomial FunctionsEssential Questions:1) How do you find the zeros of a polynomial function?

2) What is the rational zero test?

The Fundamental Theorem of Algebra:

If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the ______________________________

Linear Factorization Theorem:

If f(x) is a polynomial of degree n, where n > 0, then f has precisely n linear factors ______________________________where c1, c2, ... cn are complex numbers.

Ex. 1) Find all zeros and write the linear factorization of the function.

Please note:FTA (fundamental theorem of algebra) and LFT (linear factorization theorem) only tell us that zeros exist, not how to find them.

Methods for finding zeros:

Rational Zero TestIf the polynomial f(x) has __________ coefficients, every rational zero of f has the form:

Where _____ and _____ have no common factors other than 1 and______ = a factor of the ___________________ = a factor of the ____________­_

Ex. 2) Find the rational zeros of

Ex. 3) Find the rational zeros of

Ex. 4) Find all the real solutions of

What's the difference?Find all rational zeros:

Find all real zeros:

Find all zeros:

Conjugate PairsIf ___________ is a zero of the function, then ___________ is also a zero of the function.

Ex. 5) Find a fourth-degree polynomial with real coefficients that has 3+2i and 4-i as zeros.

Assignment:

2.5A p. 179 #7-19 (odd), 37, 39, 41