Do NowDecide whether x-1 is a factor of f(x)

Factor f(x) into linear factors given that 2 is a zero of f(x)

What is the relationship between zeroes and factors?

301732)(

7532)(

23

24

xxxxf

xxxxf

Today’s Learning Objectives

Use rational zeroes theorem to compile a list of all possible rational zeroes

Combine rational zeroes theorem with remainder theorem to test which possibilities are actual solutions

This week is the beefWe are adding beefy theorems this week that

require us to know how to use remainder theorem as a tool to test for zeroes

Must know relationship between factors and theorems

Rational Zeros TheoremGives us a method to determine all possible

candidates for rational zeros of a polynomial function

If is a rational number written in lowest terms, and if it is a zero of f, then p is a factor of the constant term and q is a factor of the leading coefficient

Key points: p/q in lowest termsp is factor of the constantq is factor of the leading coefficient

q

p

RZT- compiling the listUse RZT to compile a list of all possible

rational zeroes of the following functions:

6834)(

53193)(

128)(

23

23

23

xxxxf

xxxxf

xxxxf

RZT- what to do with the list?We have compiled the list of all possible

zeros. Now what should we do with it? What should we use it for?

Turn and talk to your neighbor and discuss some possible uses of this list; I will randomly call on a few people to share

RZT- why?RZT compiles the list of all possibilitiesSynthetic division/ remainder theorem

narrows down which ones are trueLast week: Test to see if a given value is a zero This week: Test to see if a possible value is a

zero

RZT- ex #1List all possible rational zeros of

List all p’s:List all q’s:All possible combinations (without repeats) of p/q in

lowest terms:

Find all rational zeros and FACTOR f(x) into linear factors.Use remainder theorem, looking for R=0

41127268)( 234 xxxxxf

RZT- ex #2List all possible rational zeros of

List all p’s:List all q’s:All possible combinations (without repeats) of p/q in

lowest terms:

Find all rational zeros and FACTOR f(x) into linear factors.Use remainder theorem, looking for R=0

231276)( 234 xxxxxf

RZT SummaryUse RZT to compile a list of all possibilitiesUse Remainder Theorem (with synthetic

division) to test possibilities

Seems long… how do we know when to stop? Number of Zeros Theorem:A function defined by a polynomial of degree n

has at most n distinct zeros Has as many zeros as its highest exponent

Class work- due at end of hourPage 338, section 3.3 #35-41 odd (4

problems)

Quiz at beginning of hour tomorrow: one rational zeros theorem question asking you to fully factor a polynomial

3.3 Continue HomeworkWe have not finished the section yet#8-60, multiples of 4Today’s material: 36 & 40 even