Do Now: Factor the following polynomial:

By the end of this chapter, you will be able to: - Identify all possible rational zeroes - Identify all actual zeroes - -Factor a polynomial completely - Use theorems to prove things about polynomials

What is a polynomial? A polynomial function of degree n, where n is a nonnegative integer, is defined by the form What does that mean? Descending degree of exponents, many terms

Factoring as division Factoring is a way of dividing IF AND ONLY IF what we can find perfectly divides out Example: vs. *previously, we say “cannot be factored”

Dividing without perfection With a remainder; access base knowledge of mixed number fractions

Division Algorithm Let f(x) and g(x) be polynomials with g(x) of lower degree than f(x) and g(x) with degree one or more. There are unique polynomials q(x) and r(x) such that What does this really say?

How to divide polynomials Quotient must be in the form x-k, where the coefficient on x is 1. Divisor must be written in descending order of degrees (exponents) Must use zero to represent coefficient of any missing terms Example:

Synthetic Division

Synthetic Practice

More practice Worksheet, due at end of hour

Do Now: Perform synthetic division. Write your answer in division algorithm form.

Use synthetic division to determine the remainder of a polynomial Use the remainder theorem to determine if a given value is a zero of a polynomial

Remainder Theorem If a polynomial f(x) is divided by x-k, then the remainder is equal to f(k) Prove by direct substitution

Using the remainder theorem Synthetic substitution Use the remainder theorem to find f(4) when

Testing Potential Zeroes The ZERO of a polynomial function f is a number k such that f(k)=0 ie- no remainder A zero is called a ROOT or SOLUTION Why is this important? Graphing Factoring Application problems When an object hits the ground

Testing zeroes Decide whether the given number k is a zero of f(x)

Testing zeroes: complex numbers When multiplying binomials (2 complex numbers), must FOIL or use the box

3.2 HW Due 10/10 Last chance 10/17 #20-26 even (test question) #32-38 even #42-52 even

3.2 Pop Quiz Determine whether the given value of k is a zero of the polynomial f(x). 1 2