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Agenda – Jan 61. Do Now2. Review Reading3. Notes: Rational Root
Theorem4. Work Time
Due NEXT CLASS:Rational Root Theorem
By the end of today’s class, you will be able to …• Identify and test possible roots of a polynomial function
Graphing Polynomial Functions
Get a handout please
SILENT DO NOW!
In dress codeIn your seat
Due Today: Reading – Rational Root Theorem
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 x
q
p a
q
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
pq
factors of 15
fac
tors of 1
p
q
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, 5, 5: 1p
1 3 5 15
1
pq
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, 5, 15pq
1 3 5 15
1
pq
3 2( ) 3 5 15f x x x x
Check with Factor Theorem
10x
10x 10x 10x
10x
: 1, 3, 5, 15pq
Substitute each 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, 5, 15pq
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, 5, 15pq
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
: 1, 3, 5, 15pq
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).
(15)f 3 2( ) 3( )15 15 1( 55 ) 15
( 15)f 3 2( ) 3( )15 15 1( 55 ) 15
3210 3690
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 -1513
005
1502x x
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 i 5i 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
