1

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

10

c) 4 3 210 3 84 11 60 0x x x x

11

d) 3 23 25 23 35 0x 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

26

c) 2 11

5

xy

x

d) ( )

4

xf x

x

Graph of 2

1y

x :

Characteristics:

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

31

Ex) Determine the equation of the following graphs.

a) b)

b) d)

Now Try

Page 451 #1, 3, 4, 5, 6, 7, 8, 9,

10, 13, 16, 20, 21