G10 Math Q1-Week 6- Factor Theorem.ppt

Factor Theorem &
Rational Root
Theorem
Objective:
proves the Remainder Theorem, Factor Theorem and the Rational Root Theorem.

The Factor Theorem:
 For a polynomial P(x), x – k is a factor iff P(k) =
0
iff
“if and only if”
It means that a theorem and its converse are true

If P(x) = x3
– 5x2
+ 2x + 8,
determine whether x – 4 is a factor.
4 1 -5 2 8
4 -4 -8
1 -1 -2 0
  
2 3 2
4 2 8
2 5
x x x x
x x
   

 
remainder is 0, therefore yes
other factor

Terminology:
 Solutions (or roots) of polynomial equations
 Zeros of polynomial functions
 “k is a zero of the function f if f(k) = 0”
 zeros of functions are the x values of the points
 where the graph of the function crosses the x-axis
(x-intercepts where y = 0)

Ex 1: A polynomial function and
one of its zeros are given, find
the remaining zeros:
3 2
( ) 3 4 12; 2
P x x x x
   
2 1 3 -4 -12
2 10 12
1 5 6 0
  
2
5 6 0
2 3 0
2, 3
x x
x x
x
  
  
 

Ex 2: A polynomial function and
one of its zeros are given, find
the remaining zeros:
3
( ) 7 6; 3
P x x x
   
-3 1 0 -7 6
-3 9 -6
1 -3 2 0
  
2
3 2 0
1 2 0
1, 2
x x
x x
x
  
  


Rational Root Theorem:
Suppose that a polynomial equation with
integral coefficients has the root p/q , where
p and q are relatively prime integers. Then p
must be a factor of the constant term of the
polynomial and q must be a factor of the
coefficient of the highest degree term.
(useful when solving higher degree polynomial equations)

Solve using the Rational
Root Theorem:
 4x2
+ 3x – 1 = 0 (any rational root must have a numerator
that is a factor of -1 and a denominator
that is a factor of 4)
factors of -1: ±1
factors of 4: ±1,2,4
possible rational roots: (now use synthetic division
to find rational roots)
1 1
1, ,
2 4

1 4 3 -1
4 7
4 7 6 no
-1 4 3 -1
-4 1
4 -1 0 !
yes
4 1 0
4 1
1
4
x
x
x
 


1
1,
4
x 
(note: not all possible rational roots are zeros!)

Listing Possible Rational
Roots
 When remembering how to find the list of all
possible rational roots of a polynomial,
remember the silly snake puts his tail over his
head (factors of the “tail of the polynomial”
over factors of the “head of the polynomial”).

Practice! This is how we
LEARN…

Ex 3: Solve using the Rational Root
Theorem:3 2
2 13 10 0
x x x
   
1 1 2 -13 10
1 3 -10
1 3 -10 0 !
yes
  
2
3 10 0
5 2 0
5, 2
x x
x x
x
  
  

5,1, 2
x 
1, 2, 5,10

possible rational roots:

Ex 4: Solve using the Rational Root
Theorem: 3 2
4 4 0
x x x
   
possible rational roots: 1, 2, 4

1 1 -4 -1 4
1 -3 -4
1 -3 -4 0 !
yes
  
2
3 4 0
4 1 0
1, 4
x x
x x
x
  
  

1,1, 4
x 

Ex 5: Solve using the Rational
Root Theorem:
3 2
3 5 4 4 0
x x x
   
possible rational roots:
1 2 4
1, 2, 4, , ,
3 3 3

-1 3 -5 -4 4
-3 8
3 -8 -4
-4
0 !
yes
  
2
3 8 4 0
3 2 2 0
2
, 2
3
x x
x x
x
  
  

2
1, , 2
3
x 
To find other roots can use synthetic division
using other possible roots on these coefficients.
(or factor and solve the quadratic equation)
2 3 -8 4 3 2 0
6 -4 3 2
3 -2 0
x
x
 

2
3
x 

It’s your turn
x3
+ 4x2
– 15x – 18
x – 3
x3
– 11x2
+ 14x + 80
x – 8
x3
+ 4x2
– 15x – 18
x – 3

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G10 Math Q1-Week 6- Factor Theorem.ppt

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