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Unit 4 Polynomial & Rational
Functions
Dividing Polynomials:
Review: Divide 3 22 3 12 4x x x by 2x
3 22 2 3 12 4x x x x
Most will agree that this is a painful method of division. So here
now is the new and improved method of division, Synthetic
Division.
3 22 3 12 4 2x x x x
2
Ex) Divide the following.
a) 4 3 22 5 10 4x x x x x
b) 4 22 3 5 1 3x x x x
c) 43 6 10 2x x x
d) 22 8 24 2 1x x x
3
Complete the following table:
Polynomial ( )P x
Divisor
x b
Quotient Remainder ( )P b
2 7 16x x 3x 22 3 8x x 2x
3 23 3 2x x x 1x 3 6 6x x 2x
3 22 2 3x x x 1x 3 26 17 14 2x x x 2 3x
Rule:
4
The Remainder Theorem:
When a polynomial ( )P x is divided by ( )ax b , and the
remainder is a constant, then the remainder is bPa
.
Ex) Determine the remainder when 5 36 10 17x x x is
divided by 3x .
Ex) Determine the remainder when 36 4x x is divided by
2 5x .
Ex) When the polynomial 3 2 17 6y ky y is divided by
3y the remainder is 12. What is the value of k?
5
Ex) When the polynomial 3 2( ) 3 7P x x mx nx is divided
by 2x the remainder is 3 . When it is divided by 1x
the remainder is 18 . What are the values of m and n?
Now Try
Page 124 #4, 5, 6, 7, 8, 9, 10,
12, 14, 15, C2
6
Factoring Polynomials:
The Factor Theorem:
A polynomial ( )P x has x b as a factor if and only if ( ) 0P b .
Ex) Use the factor theorem to find a factor of 3 24 17 60x x x , then use synthetic division to
completely factor it.
Integral Zero Theorem:
If x b is an integral zero of a polynomial ( )P x with integral
coefficients, then b is a factor of the constant term of the
polynomial.
Means All integers that make a polynomial equal zero are
factors of its constant term when in general form.
7
Ex) Use the integral zero theorem and the remainder theorem
to find an integral factor, then use synthetic division to
fully factor the following.
a) 3 26 19 84x x x
b) 3 26 31 4 5x x x
8
c) 4 3 24 12 3x x x x
Ex) The volume, V, of a filing cabinet can be represented by
the expression 3 22h h h , where h is the height of the
cabinet.
a) Factor the expression.
b) What are these factors representing?
c) If the height of the cabinet is 1.5m, state the other
dimensions of the cabinet.
Now Try
Page 133 #1, 3, 4, 5, 6, 7, 8,
11, 13, 15, 16,
9
Solving Polynomial Equations:
To solve polynomial equations algebraically we will use the
method of solving by factoring.
Hint: To check your answers or to help find an initial zero of
the function we can solve by graphing.
Ex) Solve the following.
a) 3 211 60 0x x x
b) 3 26 140 27x x x
12
Ex) There is a box whose width is x, height is x, and whose
length is x + 2. The volume is 45cm3. Determine the
dimensions of the box in cm.
13
Ex) Determine a polynomial equation whose roots are
1, 1 2,1 2 .
Ex) Find 3 consecutive integers with a product of 504 .
Now Try
Worksheet
14
Graphing Polynomial Functions:
Polynomial Function:
A polynomial function is a function in the form
1 2 2
1 2 2 1 0( ) ... ...n n n
n n nf x a x a x a x a x a x a
Where n is a whole number, x is a variable, and the coefficients
na to 0a are real numbers.
Ex) ( ) 2 9f x x , 4 37 9 2 11y x x x , 3 2( ) 5 2f x x x x
Ex) Indicate which of the following are polynomial functions.
a) 2( ) 6 3 7f x x x b) ( ) 7 6f x x x
c) 8 9
( )x
f xx
d) 4 24
93
y x x
e) 6 3 28 5 15
9
x x xy
f) 3 2( ) 8 2 7f x x x
g) 1
2 39 3 9y x x h) ( ) 9f x
15
The following are examples of different types of polynomials
Degree 0: Degree 1: Degree 2:
Constant Function Linear Function Quadratic Function Ex) ( ) 3f x Ex) ( ) 2 1f x x Ex) 2( ) 2 3f x x
Degree 3: Degree 4: Degree 5:
Cubic Function Quartic Function Quintic Function Ex) 3 2( ) 2 2f x x x x Ex) 4 3 2( ) 5 5 5 6f x x x x x Ex) 5 4 3 2( ) 3 5 15 4 12f x x x x x x
16
Consider the following examples of graphs of polynomial
functions:
4y x 2 1y x
3 25 8y x x x 4 22 5y x x
Odd-Degree Functions:
Even-Degree Functions:
End Behavior:
17
Zeros:
Ex) Use your calculator to find the following for the function 4 3 2( ) 3 3f x x x x x
a) zeros b) y-intercept
c) relative maximums d) domain and range
and minimums
Now Try
Page 114 #1, 2, 3, 4,
7, 10, 11,
18
Multiplicity of a Zero:
The zero of a function corresponds to the x-intercept of its
graph.
Zeros with a multiplicity of 1
Each factor of the function is unique and appears only
once.
Ex) 3 24 4 16
( 4)( 2)( 2)
y x x x
x x x
Zeros with a multiplicity of 2
When the function is factored, a factor has a multiplicity of
2 if it appears twice. The corresponding zero to that factor
is said to then have a multiplicity of 2 as well.
Ex) 3 3 2
( 2)( 1)( 1)
y x x
x x x
19
Zeros with a multiplicity of 3
When the function is factored, a factor has a multiplicity of
3 if it appears three times. The corresponding zero to that
factor is said to then have a multiplicity of 3 as well.
Ex) 3 26 12 8
( 2)( 2)( 2)
y x x x
x x x
Steps for Sketching Graphs of Polynomials:
Factor the function
Locate all x-intercepts (zeros), pay attention to any zeros
with multiplicity 2 or 3
Determine the end behavior
Draw the graph (don’t worry about exact location of
relative maximum or minimums)
20
Ex) Sketch the following graphs (exact locations of relative
maximums or minimums is not important).
a) 2 3 28y x x b) 3 27 10y x x x
c) ( 2)( 5)( 7)( 1)y x x x x d) ( 4)( 8)( 2)( 3)y x x x x
21
e) ( 5)( 2)( 7)( 2)y x x x x
f) ( 4)( 4)( 5)( 5)( 1)y x x x x x
Ex) Determine a possible functions that could describe the
following.
a) b)
22
Ex) Solve the following.
a) ( 2)( 5)( 7) 0x x x
c) ( 3)( 5)( 5)( 8) 0x x x x x
d) 3( 5)( 4)( 1) 0x x x
Now Try
Page 147 #3, 4, 5, 7, 9,
10, 13, 14, 20
23
Graph of 1
yx
:
Complete the table of values given below, then use this to sketch
the graphs of 1
yx
.
x 1y
x
10
5
1
12
110
1100
0
1100
110
12
1
5
10
Characteristics:
Non-Permissible Values
Asymptotes
24
Ex) Use your knowledge of the graph of 1
yx
and
transformations to graph the following.
a) 1
3y
x
b)
1( ) 5
2f x
x
c) 6
( )2
f xx
d) 4
35
yx
25
Consider the graph of a
y kx h
:
a
h
k
Ex) Write the following functions in the form a
y kx h
,
then sketch its graph.
a) 5
( )3
xf x
x
b)
4 5
2
xy
x
27
Ex) Use your knowledge of the graph of 2
1y
x and
transformations to graph the following.
a) 2
3
10 25y
x x
b)
2
1( )
( 3)g x
x
c) 2
53
2 1y
x x
Now Try
Page 442 #1, 2, 3, 5, 7, 8,
9, 10, 11, 15, 16
28
Graphing Rational Functions:
As we have seen rational graphs often have vertical asymptotes,
but this does not always have to be the case.
Ex) Graph the following rational functions.
a) 1
3y
x
b)
2 8 12( )
2
x xf x
x
All rational functions have restrictions (non-permissible values),
these appear on the graph as Asyptotes or Holes in the graph.
29
Tricks to Graphing Rational Functions:
When graphing rational functions it is the factors of the
numerator and denominator that tell the story.
Unique Factors of the Numerator
indicate the zeros of the function or the x-intercepts of the
graph
Unique Factors of the Denominator
indicate the restrictions of the function or the vertical
asymptotes of the graph
if the factor only appears once the graph will split (arms go
in opposite directions)
if the same factor appears twice the arms of the graph will
move in the same direction (approach or )
Factors that Appear in Both the Numerator and Denominator
indicate points of discontinuity on the graph
Horizontal Asymptotes
consider the unfactored form of the expression, the highes
power will tell the tale
if the highest power is in the numerator the graph
approaches or
if the highest power is in the denominator the graph
approaches 0 (x-axis)
if there is a tie between the numerator and denominator
consider the coefficients
30
Ex) Graph the following rational functions.
a) 1
5y
x
b)
2
1( )
6f x
x x
c) 2 6 8
( )4
x xf x
x
d)
2
2
10 21( )
12
x xf x
x x