Do Now:Factor the following polynomial:

12194

62

2

2

xx

xx

Chapter 3: Polynomial Theory

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

011

1 ...)( axaxaxaxf nn

nn

Factoring as divisionFactoring is a way of dividing IF AND ONLY

IF what we can find perfectly divides outExample:

vs.

*previously, we say “cannot be factored”

62 2 xx 62 2 xx

Dividing without perfectionWith a remainder; access base knowledge of

mixed number fractions

Division AlgorithmLet 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?

)()()()( xrxqxgxf

)(

)()(

)(

)(

xq

xrxg

xq

xf

How to divide polynomialsQuotient 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 termsExample:

Synthetic Division

4

15023 23

x

xx

Synthetic Practice

2

22865 23

x

xxx

3

1011154 23

x

xxx

More practiceWorksheet, due at end of hour

Do Now:Perform synthetic division. Write your answer

in division algorithm form.

2

22865 23

x

xxx

Today’s Learning Targets: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 TheoremIf a polynomial f(x) is divided by x-k, then the

remainder is equal to f(k)Prove by direct substitution

2- k when 22865)( 23 xxxxf

2

22865 23

x

xxx

Using the remainder theoremSynthetic substitutionUse the remainder theorem to find f(4) when

2550153)( 34 xxxxf

Testing Potential ZeroesThe 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 SOLUTIONWhy is this important?

GraphingFactoringApplication problems

When an object hits the ground

Testing zeroesDecide whether the given number k is a zero

of f(x)2 k when 1832)( 23 xxxf

3- k when 4536144)( 234 xxxxxf

Testing zeroes: complex numbersWhen multiplying binomials (2 complex

numbers), must FOIL or use the box

2i1 k when 5242)( 234 xxxxxf

3i1 k when 20146)( 234 xxxxxf

3.2 HWDue 10/10Last chance 10/17

#20-26 even (test question)#32-38 even#42-52 even

3.2 Pop QuizDetermine whether the given value of k is a

zero of the polynomial f(x).

1

2

0 k when 107)( 23 xxxxf

2k when 443)( 23 xxxxf