Today’s Agenda – Feb 31. Silent Do Now

PRE-CALCULUS 1

2. NotesRational Root Theorem

Due TODAY: HW #51 Graphing Polynomials - ReduxDue TOMORROW: HW #52Rational Root

Theorem

3. Work Time

SWBAT…• Apply rational root theorem to identify possible roots of a polynomial• Use synthetic/long division or factor theorem to test possible roots

4. Questions/Summary

FACTOR THIS…10x

10x 10x 10x

10x

3 2( ) 3 5 15f x x x x Without more information, our best hope

of factoring this cubic function is to guess and check.

There is a better way…

The Rational Root Theorem is a tool for predicting the values of Rational Roots. The theorem says:

10x

10x 10x 10x

10x

The Rational Root Theorem

10 1 1If ( ) ... ,

where the coeffiecients are all integers,

and a rational zero of ( ) in reduced form is ,

then must be a factor of (the constant term) must be a factor

n nn n

n

P x a x ax a x a

pP xq

p aq

0of (the leading coefficient).a

Apply theRational Root Theorem

10x

10x 10x 10x

10x

3 2( ) 3 5 15f x x x x Are the coefficients all integers?

Then possible rational roots are I

where

pqfactors of 15

fac tors of 1

pq

3 2( ) 31 155f x x x x

3 2( ) 31 155f x x x x

Apply theRational Root Theorem

10x

10x 10x 10x

10x

Factors of constant term, Factors of leading coefficient ,All possible rational roots of the form are: I

1, 3,: 5p

1 3 5 1pq

1: q pq

3 2( ) 3 5 15f x x x x

Apply theRational Root Theorem

10x

10x 10x 10x

10x

So the possible zeros are:I: 1, 3, 5pq

1 3 5 1pq

3 2( ) 3 5 15f x x x x

Check with Factor Theorem

10x

10x 10x 10x

10x

: 1, 3, 5pq

Substitute each of our possible rational roots into f(x). If the value is a root, then f(value) = 0

( )1f 3 2( ) 3( ) 5( ) 11 1 1 5

1( )f 3 2( ) 3( ) 5( )1 1 51 1 12

24

3 2( ) 3 5 15f x x x x

Check with Factor Theorem

10x

10x 10x 10x

10x

: 1, 3, 5pq

Continue testing possiblerational roots for f(x).

( )3f 3 2( ) 3( ) 5( ) 13 3 3 5

3( )f 3 2( ) 3( ) 5( )3 3 53 1 0

84

: 1, 3, 5pq

3 2( ) 3 5 15f x x x x

Check with Factor Theorem

10x

10x 10x 10x

10x

Continue testing possiblerational roots for f(x).

( )5f 3 2( ) 3( ) 5( ) 15 5 5 5

5( )f 3 2( ) 3( ) 5( )5 5 55 1 60

240

3 2( ) 3 5 15f x x x x

Now what?10x

10x 10x 10x

10x

so must be a factor of f(x).( 03)f ( 3)xUse long or synthetic division to further factor

f(x) 3 1 -3 5 -15130

05

150

3 2( ) 3 5 15f x x x x

Rewrite in Factored Form

10x

10x 10x 10x

10x

( )f x ( 3)x 2( 0 5)x x Simplify!( )f x ( 3)x 2( 5)x

Simplify further!

3 2( ) 3 5 15f x x x x

Rewrite in Factored Form

10x

10x 10x 10x

10x

( )f x ( 3)x 2( 5)x 2 5x 0

2 5x 5x 5x i

( )x

3 2( ) 3 5 15f x x x x

Rewrite in Factored Form

10x

10x 10x 10x

10x

( )f x ( 3)x 2( 5)x

5x i5i 5i

( )f x ( 3)x ( )x

The Rational Root Theorem says:

10x

10x 10x 10x

10x

The Rational Root Theorem

10 1 1 ( ) ... ,

where the coeffiecients are all integers,

positive factors of constant term positive factors of leading coefficien

If

then

t

n nn nP x a x ax a x a

Rational Zero

Key Point