unit 4 polynomial & rational functions1 unit 4 polynomial & rational functions general...
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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− +
=− +