Unit 4 Polynomial & Rational Functions1 Unit 4 Polynomial & Rational Functions General...

1

Unit 4 Polynomial & Rational

Functions

General Outcome: • Develop algebraic and graphical reasoning through the study of relations.

Specific Outcomes:

4.1 Demonstrate an understanding of factoring polynomials of degree greater than 2 (limited

to polynomials of degree 5 with integral coefficients).

4.2 Graph and analyze polynomial functions (limited to polynomial functions of degree 5).

4.3 Graph and analyze rational functions (limited to numerators and denominators that are

monomials, binomials, or trinomials).

Topics:

• Dividing Polynomials / Remainder Theorem (Outcome 4.1)

• Factoring Polynomials (Outcome 4.1)

• Solving Polynomial Equations (Outcome 4.1)

• Graphing Polynomial Functions (Outcome 4.2)

• Multiplicity of a Zero (Outcome 4.2)

• Graphs of 1

yx

= and 2

1y

x= (Outcome 4.3)

• Graphing Rational Functions (Outcome 4.3)

2

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+ − + −

3

Alternate Forms of Division:

Synthetic Division

( ) ( )3 22 3 12 4 2x x x x− + − +

Area Model of Division

( ) ( )3 22 3 12 4 2x x x x− + − +

4

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+ + −

5

Complete the following table:

Polynomial ( )P x

Divisor

x b−

Quotient Remainder ( )Pb

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:

6

The Remainder Theorem:

When a polynomial ( )P x is divided by ( )ax b− , and the

remainder is a constant, then the remainder is ( )bP a .

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?

7

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?

8

Dividing Polynomials & the Remainder Theorem Assignment:

1) Perform each division. Express the result in the form ( )

( )P x R

Q xax b ax b

= +− −

.

Identify any restrictions on the variable.

a) ( ) ( )3 2 3 4x x x+ + + b) 4 3 22 12 6

2m m m m

m− + + −

−

c) 3 22 7 8

2 3x x x

x+ − −

+ d) ( ) ( )3 27 3 4 2x x x x+ − + +

e) 411 4 73

t tt− −−

f) 3 22 3 9

3h h h

h+ − +

+

9

g) ( ) ( )3 26 4 1 5x x x x+ − + − h) ( ) ( )34 15 2 3n n n− + +

i) 5 4 2

2

4 113

x x xx

− + +−

j) 4 2

2

3 5 63 1

x xx+ +−

2) Determine the remainder when each division is performed.

a) 3 23 5 2

2x x x

x+ − +

+ b)

3 22 3 93

x x xx

+ − ++

10

c) ( ) ( )3 22 3 5 5x x x x+ − + − d) ( ) ( )3 24 6 3 2 5n n n− + −

3) When 3 24x x x k+ − + is divided by 1x− the remainder is 16. Determine the

value of k.

4) When 3 2 5n kn n+ + + is divided by 2n+ the remainder is 3. Determine the

value of k.

5) For what value of c will the polynomial 3 2( ) 2 5 2P x x cx x=− + − + have the

same remainder when it is divided by 2x− and by 1x+ ?

11

6) When 23 6 10x x+ − is divided by x k+ the remainder is 14. Determine the

value(s) of k.

7) When the polynomial 3 23 9x ax bx+ + − is divided by 2x− , the remainder

is 5− . When it is divided by 1x+ , the remainder is 16− . What are the values

of a and b?

12

Factoring Polynomials:

The Factor Theorem:

A polynomial ( )P x has x b− as a factor if and only if ( ) 0Pb = .

Ex) Use the factor theorem to find a factor of 3 24 17 60x x x− − + , then use this 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.

13

Ex) Use the integral zero theorem and the remainder theorem

to find an integral factor, then use this to fully factor the

following.

a) 3 26 19 84x x x− − +

b) 3 26 31 4 5x x x+ + −

14

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.

15

Factoring Polynomials Assignment:

1) Determine the corresponding binomial factor of a polynomial ( )P x , given the

value of the zero.

a) (1) 0P = b) ( 3) 0P− = c) ( ) 0P a =

2) Determine whether 1x− is a factor of the following polynomials.

a) 3 23 4 2x x x− + − b) 3 22 3 2x x x− − −

3) Determine whether 2x+ is a factor of the following polynomials.

a) 25 2 6x x+ + b) 3 22 5 8x x x− − −

4) Factor each of the following polynomials fully.

a) 3 2( ) 6 11 6P x x x x= − + −

16

b) 3 2( ) 2 2P x x x x= + − −

c) 3 2( ) 16 16Pt t t t= + − −

d) 3 27 10h h− +

17

e) 5 4 3 23 5 15 4 12k k k k k+ − − + +

f) 4 3 24 7 34 24h h h h+ − − −

g) 4 3 22 2 2 3x x x x+ + − −

18

5) Determine the value(s) of k so that the binomial is a factor of the polynomial.

a) 2x x k− + , 2x− b) 2 16x kx+ − , 2x−

6) The volume, ( )V h , of a bookcase can be represented by the expression 3 22h h h− + , where h is the height of the bookcase. What are the possible

dimensions of the bookcase in terms of h?

7) The volume of water in a rectangular fish tank can be modeled by the

polynomial 3 2( ) 14 63 90V x x x x= + + + . If the depth of the tank is given by the

polynomial 6x+ , what polynomials represent the possible length and width of

the fish tank?

19

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+ = +

20

c) 4 3 210 3 84 11 60 0x x x x− − − + =

21

d) 3 23 25 23 35 0x x x+ + − =

22

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.

23

Ex) Determine a polynomial equation whose roots are

1,1 2,1 2+ − .

Ex) Find 3 consecutive integers with a product of 504− .

24

Solving Polynomial Equations Assignment:

1) Solve the following.

a) ( 1)( 4)( 5) 0x x x+ − + = b) ( 2)( 7)( 6) 0y y y− − + =

c) ( 3)( 8) 0x x x+ − = d) 2( 6)( 3) 0x x+ − =

2) Solve the following. Leave answers as exact roots if necessary.

a) 3 27 12 0x x x+ + = b) 3 25 12 36x x x− = −

25

c) 3 24 6 0x x x− + + = d) (4 1)(3 1)( 1) 0x x x+ − + =

e) 3 10 3 0x x− + = f) 3 25 7 8 4 0x x x− − + =

g) 3 24 8 5 10 0x x x− − + = h)

3 23 4 12 0x x x− − + =

26

i) 3 22 13 16 5 0x x x− + − = j) 2 37 5 3x x x− = −

k) 3 216 8 0x x x+ + = l) 3 25 11 17 15 0b b b− − + =

m) 39 4 0x x− = n) 3 26 29 12 23x x x+ − =

27

o) ( 4)( 1) 4x x x+ + = p) 3 2 16 20 0x x x− − − =

q) 3 218 15 4 4 0x x x+ − − = r) 3 23 8 1x x+ =

3) Solve the following.

a) 2

3 133

4x

x x− + =− b) 2 1

2 1x

x x− =

− −

28

4) One root of each equation is 2− . Evaluate k and find the other roots.

a) 3 2 10 24 0x kx x+ − − = b) 3 23 4 2 0x x kx+ + − =

5) Solve each of the following.

a) 4 3 24 6 0x x x x− + + =

29

b) 4 24 3 0x x− + =

c) 4 3 22 5 3 1 0x x x x+ + − − =

30

6) A toothpaste box has square ends. The length is 12 cm greater than the width.

The volume of the box is 135 cm3. What are the dimensions of the box?

7) Solve the following polynomial equation.

5 4 3 23 5 15 4 12 0x x x x x+ − − + + =

31

Graphing Polynomial Functions:

Polynomial Functions:

A polynomial function is a function in the form

1 2 2

1 2 2 1 0( ) ... ...n n nn n nf x a x a x a x a x ax 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) 4 249

3y x x= −

e) 6 3 28 5 15

9x x x

y− + −

= f) 3 2( ) 8 2 7f x x x−= + +

g) 12 39 3 9y x x= − + h) ( ) 9f x =

32

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= + − − + +

33

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:

34

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

35

Graphing Polynomial Functions Assignment:

1) Identify whether each of the following is a polynomial function.

a) ( ) 2h x x= − b) 3 1y x= + c) ( ) 3xf x =

d) 4( ) 3 7g x x= − e) 3 2( ) 3p x x x x−= + + f) 24 2 5y x x=− + +

2) State the degree, type, leading coefficient, and constant term for each of the

following polynomial functions.

a) ( ) 3f x x=− + b) 4 2( ) 3 3 2 1g x x x x= + − +

Degree: Degree:

Type: Type:

Leading Coefficient: Leading Coefficient:

Constant Term: Constant Term:

c) 3( ) 4 3k x x= − d) ( ) 6h x =−

Degree: Degree:

Type: Type:

Leading Coefficient: Leading Coefficient:

Constant Term: Constant Term:

3) Determine the following for each of the following; whether the graph

represents an odd-degree or even degree polynomial function, whether the

leading coefficient is positive or negative, the number of x-intercepts, the

domain and range.

a)

Odd / Even Polynomial

Function:

Positive / Negative

Leading Coefficient:

Number of x-intercepts:

Domain:

Range:

36

b)

Odd / Even Polynomial

Function:

Positive / Negative

Leading Coefficient:

Number of x-intercepts:

Domain:

Range:

c)

Odd / Even Polynomial

Function:

Positive / Negative

Leading Coefficient:

Number of x-intercepts:

Domain:

Range:

d)

Odd / Even Polynomial

Function:

Positive / Negative

Leading Coefficient:

Number of x-intercepts:

Domain:

Range:

37

4) For each of the following polynomial functions determine the sign of the

leading coefficient, the far right end behavior, the possible number of

x-intercepts, and the value of the y-intercept.

a) 2( ) 3 1f x x x= + − b) 3 2( ) 4 2 5g x x x x=− + − +

Sign of Leading Sign of Leading

Coefficient: Coefficient:

Far Right End Behavior: Far Right End Behavior:

Possible Number of Possible Number of

x-intercepts: x-intercepts:

Value of y-intercept: Value of y-intercept:

c) 4 3 2( ) 7 2 3 6 4h x x x x x=− + − + + d) 5 2( ) 3 9q x x x x= − +

Sign of Leading Sign of Leading

Coefficient: Coefficient:

Far Right End Behavior: Far Right End Behavior:

Possible Number of Possible Number of

x-intercepts: x-intercepts:

Value of y-intercept: Value of y-intercept:

e) ( ) 4 2p x x= − f) 3 4 2( ) 2 4v x x x x=− + −

Sign of Leading Sign of Leading

Coefficient: Coefficient:

Far Right End Behavior: Far Right End Behavior:

Possible Number of Possible Number of

x-intercepts: x-intercepts:

Value of y-intercept: Value of y-intercept:

38

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= − += + − −

39

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)

40

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=− + − + −

41

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)

42

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− + − +

43

Multiplicity of a Zero Assignment:

1) Use the graph of the given polynomial functions to write the corresponding

equation. Assume all x and y-intercepts are integer values.

a)

b)

c)

44

2) For each polynomial graph determine the x-intercepts (assume these are integer

values), the intervals where the function in positive, the intervals where the

function is negative, and list all factors of the function that have a multiplicity

of 1, 2 and 3.

a)

x-intercepts:

Positive Intervals:

Negative Intervals:

Factor(s) of Multiplicity 1:

Factor(s) of Multiplicity 2:

Factor(s) of Multiplicity 3:

b)

x-intercepts:

Positive Intervals:

Negative Intervals:

Factor(s) of Multiplicity 1:

Factor(s) of Multiplicity 2:

Factor(s) of Multiplicity 3:

c)

x-intercepts:

Positive Intervals:

Negative Intervals:

Factor(s) of Multiplicity 1:

Factor(s) of Multiplicity 2:

Factor(s) of Multiplicity 3:

45

d)

x-intercepts:

Positive Intervals:

Negative Intervals:

Factor(s) of Multiplicity 1:

Factor(s) of Multiplicity 2:

Factor(s) of Multiplicity 3:

3) Without using technology, match each function with its corresponding graph.

i) ( )42( 1) 2y x= − − ii) 3( 2) 2y x= − −

iii) 40.5 3y x= + iv)

3( ) 1y x= − +

a) b)

c) d)

46

4) Without using technology, sketch the graph of each function. Factor the

function where necessary and label all intercepts.

a) 3 24 45y x x x= − −

b) 4 2( ) 81f x x x= −

c) 3 2( ) 3 3h x x x x= + − −

47

d) 4 3 2( ) 4 6f x x x x x= − + +

e) 3 25 7 3y x x x=− + − +

f) 2 2( ) ( 1)( 1) ( 3)g x x x x= − + +

g) 2 3( ) ( 3) ( 2)h x x x=− + −

48

5) Determine the equation of each polynomial function with the following

characteristics.

a) a cubic function with zeros of 3− (multiplicity 2) and 2, and a y-intercept

of 18−

b) a quantic function with zeros of 1− (multiplicity 3) and 2 (multiplicity 2),

and a y-intercept of 20

c) a quartic function with zeros of 2− (multiplicity 2) and 3 (multiplicity 2),

and a constant term of 6−

d) a cubic function with x-intercepts of 3, 3− and 1, and goes through the

point ( )4, 91− .

49

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

50

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

= −+

51

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

2x

yx−

=−

52

c) 2 11

5x

yx−

=+

d) ( )4

xf x

x=

−

Graph of 2

1y

x= :

Characteristics:

53

Ex) Use your knowledge of the graph of 2

1y

x= and

transformations to graph the following.

a) 2

310 25

yx x

=− +

b) 2

1( )

( 3)g x

x−

=−

c) 2

53

2 1y

x x= −

+ +

54

Graphs of 1

yx

= and 2

1y

x= Assignment:

1) Without using technology, match each equation with its corresponding graph.

i) 2

1yx

= − ii) 2

1y

x=

+ iii)

21

yx

=−

iv) 2

1yx

= +

a) b)

c) d)

55

2) Without using technology, sketch the graph of each of the following functions.

Identify the domain, range, intercepts and asymptotes .

a) 1

2y

x=

+

Domain:

Range:

Intercepts:

Asymptotes:

b) 2

1( 4)

yx

=−

Domain:

Range:

Intercepts:

Asymptotes:

56

c) 4

3yx

= +

Domain:

Range:

Intercepts:

Asymptotes:

d) 2

35

( 4)y

x−

= +−

Domain:

Range:

Intercepts:

Asymptotes:

e) 8

xy

x=

+

Domain:

Range:

Intercepts:

Asymptotes:

57

f) 26

xy

x− −

=+

Domain:

Range:

Intercepts:

Asymptotes:

3) Determine the equation of each function in the form a

y kx h

= +−

.

a) b)

c) d)

58

4) The rational function 7

ay k

x= +

− passes through the point ( )10, 1 and ( )2, 9 .

Determine the values of a and k.

5) Write the function 2

2 4x

yx−

=+

in the form a

y kx h

= +−

and then sketch its

graph.

59

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( )

2x x

f xx+ +

=+

All rational functions have restrictions (non-permissible values),

these appear on the graph as Vertical Asyptotes or Points of

Discontinuity (holes) in the graph.

60

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

61

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( )

12x x

f xx x+ +

=− −

62

Ex) Determine the equation of the following graphs.

a) b)

b) d)

63

Graphing Rational Functions Assignment:

1) Given the graph of 2

46 8

xy

x x−

=− +

, determine its characteristics.

Non-Permissible value(s):

Location of Asymptotes:

Location of Point of Discontinuity:

Domain:

Range:

2) Explain any differences in the graphs of 2 2 3

( )3

x xf x

x− −

=+

and

2 2 3g( )

3x x

xx+ −

=+

.

3) For each function, determine the locations of any vertical asymptotes,

horizontal asymptotes, points of discontinuity, x-intercepts, and y-intercept.

a) 2

2

49 20

x xy

x x+

=+ +

Vertical Asymptote(s):

Horizontal Asymptote:

Point(s) of Discontinuity:

x-intercept(s):

y-intercept:

64

b) 2

2

2 5 31

x xy

x− −

=−

Vertical Asymptote9s):

Horizontal Asymptote:

Point(s) of Discontinuity:

x-intercept(s):

y-intercept:

c) 2

2

2 82 8

x xy

x x+ −

=− −

Vertical Asymptote(s):

Horizontal Asymptote:

Point(s) of Discontinuity:

x-intercept(s):

y-intercept:

d) 2

2

2 7 159 4

x xy

x+ −

=−

Vertical Asymptote(s):

Horizontal Asymptote:

Point(s) of Discontinuity:

x-intercept(s):

y-intercept:

65

4) Without using technology, match each rational function with its graph.

i) 2

2

24

x xy

x+

=+

ii) 2

22

xy

x x−

=−

iii) 2

24

xy

x+

=−

iv) 2

22

xy

x x=

+

a) b)

c) d)

5) Determine the equation for each rational function below.

a) b)

66

6) Without using technology, match each rational function with its graph.

i) 2

2

2( )

20x x

f xx x+ −

=+ −

ii) 2

2

5 4( )

2x x

g xx x− +

=− −

iii) 2

2

5 6( )

5 4x x

h xx− +

=− +

iv) 2

2

12( )

3 10x x

j xx x+ −

=− −

a) b)

c) d)

7) Explain why the graphs of 2

3( )

5 6x

f xx x

−=

− − and 2

3( )

5 6x

g xx x

−=

− + are so

different from one another.

67

8) For each case below, determine the equation of a possible rational function

with the characteristics given.

a) vertical asymptotes at 5x= and x-intercepts of 10−

b) a vertical asymptote at 4x=− , a point of discontinuity at 11

, 92−

, and

an x-intercept of 8

c) a point of discontinuity at 1

2, 5

−

, a vertical asymptote at 3x= and an

x-intercept of 1−

d) vertical asymptote at 3x= and 67

x= , and x-intercepts of 14−

and 0

68

9) Determine the equations for the rational functions shown below.

a) b)

10) Given 2

2

2 4( )

3 28x x

f xx x

−=

+ −, determine the equation of ( )1

( 3)4

y f x= − − in

simplest form.

11) Determine the equation of the rational functions shown below.

a) b)

69

Answers:

Dividing Polynomials / Remainder Theorem:

1. a) 3 2

23 453 12

4 4x x

x xx x+ +

= − + −+ +

b) 4 2

3 22 3 5 1 1212 6 15 40

3 3x x x

x x xx x

− + += − + − +

+ +

c) 3 2

22 7 8 22

2 3 2 3x x x

x xx x

+ − −= − − −

+ +

d) 3 2

27 3 4 349 15

2 2x x x

x xx x

+ − += + + +

− −

e) 4

3 211 4 7 2984 12 36

3 3t t

t t tt t− −

=− − − −− −

f) 3 2

22 3 9 93 3

h h hh h

h h+ − +

= − ++ +

g) 3 2

26 4 1 25611 51

5 5x x x

x xx x

+ − += + + +

− −

h) 3

24 15 2 614 12 21

3 3n n

n n nn n− +

= − + −+ +

i) 5 4 2

3 22 2

4 11 9 224 3 11

3 3x x x x

x x xx x

− + + −= − + − +

− −

j) 4 2

22 2

3 5 6 82

3 1 3 1x x

xx x+ +

= + +− −

2. a) 16 b) 9 c) 165 d) 28

3. 12

4. 2

5. 11

6. 2, 4k=−

7. 23

a−

= , 143

b−

=

70

Factoring Polynomials Assignment:

1. a) 1x− b) 3x+ c) x a−

2. a) yes b) no

3. a) no b) no

4. a) ( 1)( 2)( 3)x x x− − − b) ( 1)( 1)( 2)x x x− + + c) ( 1)( 4)( 4)x x x+ − +

d) ( )2( 5) 5 2x x x− + − e) ( 2)( 1)( 1)( 2)( 3)x x x x x− − + + +

f) ( 3)( 1)( 2)( 4)x x x x− + + + g) ( )2( 1)( 1) 2 3x x x x− + + +

5. a) 2k=− b) 6k=

6. ( 1) ( 1)h h h − −

7. 3x+ and 5x+

Solving Polynomial Equations Assignment:

1. a) 5, 1, 4x=− − b) 6, 2, 7y=− c) 3, 0, 8x=− d) 6, 3x=−

2. a) 4, 3, 0x=− − b) 3, 2, 6x=− c) 1, 2, 3x=−

d) 1 1

1, , 4 3

x−

=− e) 3 13

3, 2

x−

= f) 2

1, , 25

x=−

g) 5 5

, , 22 2

x−

= h) 2, 2, 3x=− i) 1

, 1, 52

x= j) 1, 3x=

k) 0x= l) 2 29

3, 5

b−

= m) 2 2, 0,

3 3x

−= n)

43

x=

o) 3 17

2, 2

x−

=− p) 2, 5x=− q) 2 1,

3 2−

r) 1 3 5,

3 2−

3. a) 3, 2

4x

−= b)

3 50,

2x

=

4. a) 4− and 3 b) 13−

and 1

5. a) 1, 0, 2, 3x=− b) 3, 1, 1, 3x=− − c) 1

1, 2

x=−

6. 3cm 3 cm 15 cm

7. 3, 2, 1, 1, 2x=− − −

Graphing Polynomial Functions Assignment:

1. a) no b) yes c) no d) yes e) no f) yes

71

2. a) Degree: 1 b) Degree: 4

Type: Linear Type: Quartic

Leading Coefficient: 1− Leading Coefficient: 3

Constant Term: 3 Constant Term: 1

c) Degree: 3 b) Degree: 0

Type: Cubic Type: Constant

Leading Coefficient: 3− Leading Coefficient: 6−

Constant Term: 4 Constant Term: 6−

3. a) Odd / Even Polynomial b) Odd / Even Polynomial

Function: Odd Function: Odd

Positive / Negative Positive / Negative

Leading Coefficient: Positive Leading Coefficient: Positive

Number of x-intercepts: 3 Number of x-intercepts: 5

Domain: x R Domain: x R

Range: y R Range: y R

c) Odd / Even Polynomial d) Odd / Even Polynomial

Function: Even Function: Even

Positive / Negative Positive / Negative

Leading Coefficient: Negative Leading Coefficient: Negative

Number of x-intercepts: 3 Number of x-intercepts: None

Domain: x R Domain: x R

Range: 16.9y Range: 3y−

4. a) Sign of Leading b) Sign of Leading

Coefficient: Positive Coefficient: Negative

Far Right End Behavior: Up Far Right End Behavior: Down

Possible Number of Possible Number of

x-intercepts: 2 x-intercepts: 3

Value of y-intercept: 1− Value of y-intercept: 5

c) Sign of Leading d) Sign of Leading

Coefficient: Negative Coefficient: Positive

Far Right End Behavior: Down Far Right End Behavior: Up

Possible Number of Possible Number of

x-intercepts: 4 x-intercepts: 5

Value of y-intercept: 4 Value of y-intercept: 0

72

e) Sign of Leading f) Sign of Leading

Coefficient: Negative Coefficient: Negative

Far Right End Behavior: Down Far Right End Behavior: Down

Possible Number of Possible Number of

x-intercepts: 1 x-intercepts: 3

Value of y-intercept: 4 Value of y-intercept: 0

Multiplicity of a Zero Assignment:

1. a) ( 3)( 2)( 1)y x x x= + + − b) ( 4)( 1)( 3)y x x x=− + − −

c) 2( 4) ( 1)( 3)x x x− + − −

2. a) x-intercepts: 4, 1, 1− − b) x-intercepts: 1, 4−

Positive Intervals: ( ) ( )4, 1 1,− − Positive Intervals: None

Negative Intervals: ( ) ( ), 4 1,1−− − Negative Intervals: , 1,4x R x −

Factor(s) of Multiplicity 1: ( 4),x+ Factor(s) of Multiplicity 1: None

( 1),( 1)x x+ − Factor(s) of Multiplicity 2: ( 1),x+

Factor(s) of Multiplicity 2: None ( 4)x−

Factor(s) of Multiplicity 3: None Factor(s) of Multiplicity 3: None

c) x-intercepts: 3, 1− d) x-intercepts: 1, 3−

Positive Intervals: ( ) ( ), 3 1,−− Positive Intervals:( ), 1−−

Negative Intervals: ( )3,1− Negative Intervals: ( ) ( )1,3 3,−

Factor(s) of Multiplicity 1: ( 3)x+ Factor(s) of Multiplicity 1: None

Factor(s) of Multiplicity 2: None Factor(s) of Multiplicity 2: ( 3)x−

Factor(s) of Multiplicity 3: ( 1)x− Factor(s) of Multiplicity 3: ( 1)x+

3. a) ii b) iv c) iii d) i

4. a) ( 9)( 5)y x x x= − + b) 2( ) ( 3)( 3)f x x x x= − +

73

c) ( ) ( 1)( 1)( 3)h x x x x= − + + d) ( ) ( 3)( 2)( 1)f x x x x x= − − +

e) 2( 3)( 1)y x x= − − f) 2 2( ) ( 1)( 1) ( 3)g x x x x= − + +

g) 2 3( ) ( 3) ( 2)h x x x=− + −

5) a) 2( 3) ( 2)y x x= + − b) 3 25( 1) ( 2)y x x= + − c) 2 21

( 2) ( 3)6

y x x−

= + −

d) ( )273 ( 1)

3y x x

−= − −

Graphs of 1

yx

= and 2

1y

x= Assignment:

1. a) ii b) i c) iv d) iii

2. a)

Domain: 2,xx x R−

Range: 0,y y y R

Intercepts: None

Asymptotes: 2x=− , 0y=

74

b)

Domain: 4,x x x R

Range: 0,y y y R

Intercepts: 1

0, 16

Asymptotes: 4x= , 0y=

c)

Domain: 0,x x x R

Range: 3,y y y R

Intercepts: 4

, 03−

Asymptotes: 0x= , 3y=

d)

Domain: 4,x x x R

Range: 5,y y y R

Intercepts: ( )3.225, 0 & ( )4.775, 0

Asymptotes: 4x= , 5y=

e)

Domain: 8,xx x R−

Range: 1,y y y R

Intercepts: ( )0, 0

Asymptotes: 8x=− , 1y=

f)

Domain: 6,xx x R−

Range: 1,y y y R−

Intercepts: ( )2, 0−

Asymptotes: 6x=− , 1y=−

75

3. a) 4

yx−

= b) 1

3y

x=

+ c)

84

2y

x= +

− d)

46

1y

x−

= −−

4. 24a= , 7k=−

5. 2 12 2

yx−

= ++

Graphing Rational Functions Assignment:

1. Non-Permissible value(s): 2, 4x , Location of Asymptotes: 2x= , Location

of Point of Discontinuity: 1

4, 2

, Domain: 2,4,xx x R ,

Range: 1

0, ,2

y y y R

2. Although the graphs of ( )f x and ( )g x both have non-permissible values of

3x− , the graph of ( )f x has a vertical asymptote at 3x=− whereas the

graph of ( )g x has a point of discontinuity at ( )3, 4− − .

3. a) Vertical Asymptote(s): 5x=− , Horizontal Asymptote: 1y= , Point(s) of

Discontinuity: ( )4, 4− − , x-intercept(s): 0, y-intercept: 0

b) Vertical Asymptote(s): 1x=− , 1x= , Horizontal Asymptote: 2y= ,

Point(s) of Discontinuity: None, x-intercept(s): 12−

, 3, y-intercept: 3

c) Vertical Asymptote(s): , 4x= , Horizontal Asymptote: 1y= , Point(s) of

Discontinuity: None, x-intercept(s): 4− , 2, y-intercept: 1

d) Vertical Asymptote(s): 32

x−

= , Horizontal Asymptote: 12

y−

= , Point(s)

of Discontinuity: 3 13,

2 12−

, x-intercept(s): 5− , y-intercept: 53−

4. a) ii b) iv c) i d) iii

5. a) ( 6)( 2)

x xy

x x+

=+

b) ( 3)( 7)( 3)( 1)x x

yx x

− + −=

+ −

6. a) iii b) ii c) iv d) i

76

7. Because 2

3 3( )

5 6 ( 6)( 1)x x

f xx x x x

− −= =

− − − + its graph will have vertical

asymptotes at 6x= and 1x=− and an x-intercept of 3. Although the equation

of ( )g x initially looks very similar to the equation of ( )f x , once factored it is

quite different, 2

3 3( )

5 6 ( 3)( 2)x x

g xx x x x

− −= =

− + − −. This graph will have a

vertical asymptote at 2x= and a point of discontinuity when 3x= . It does not

have an x-intercept.

8. a) 10

( 5)( 5)x

yx x

+=

− + b)

(2 11)( 8)(2 11)( 4)

x xy

x x+ −

=+ +

c) ( 2)( 1)( 2)( 3)x x

yx x+ +

=+ −

d) (4 1)

( 3)(7 6)x x

yx x

+=

− −

9. a) 3( 3)( 2)( 3)( 2)

x xy

x x− + −

=+ −

b) ( 6)( 2)( 2)( 3)

x xy

x x− + −

=+ −

10. ( 3)( 1)

2( 10)( 1)x x

yx x− −

=− +

11. a) (3 4)( 2)( 4)

4( 2)( 4)x x x

yx x+ − +

=− +

b) 2( 1)( 2)( 1)( 2)

x x xy

x x− +

=− +