Presentation on theme: "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."— Presentation transcript:
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:
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