Finding factors of polynomial functions

1. Factor Theorem & Rational
Root Theorem
Objective:
SWBAT find zeros of a polynomial by
using Rational Root Theorem
(also known as Rational Zeros Theorem)

2. 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

3. 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

4. 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)

5. 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
  
  
  

6. 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
  
  


7. 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)

8. 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!)

9. 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”).

10. 
Practice! This is how we LEARN…

11. 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:

12. 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  

13. 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 

14. Section 3.3 – Polynomial Functions
Definition:
The graph of the function
touches the x-axis but does
not cross it.
Zero Multiplicity of an Even Number
Multiplicity
The number of times a factor (m) of a function is
repeated is referred to its multiplicity (zero multiplicity
of m).
The graph of the function
crosses the x-axis.
Zero Multiplicity of an Odd
Number

15. Section 3.3 – Polynomial Functions
3 is a zero with a multiplicity of
Identify the zeros and their multiplicity
3
-2 is a zero with a multiplicity of
1 Graph crosses the x-axis.
Graph crosses the x-axis.
-4 is a zero with a multiplicity of
2
7 is a zero with a multiplicity of
1 Graph crosses the x-axis.
Graph touches the x-axis.
-1 is a zero with a multiplicity of
1
4 is a zero with a multiplicity of
1 Graph crosses the x-axis.
Graph crosses the x-axis.
2
2 is a zero with a multiplicity of Graph touches the x-axis.