Deck 6: Rational Expressions, Functions, and Equations

Question 1

Find the function value.
$$f ( x ) = \frac { x - 4 } { 7 x + 12 } ; f ( - 2 )$$

A) 0
B) 1
C) $- 3$
D) 3

Answer 1

D

Question 2

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.
Find f(8).

A) 3
B) 8
C) -10.8
D) -8

Answer 2

D

Question 3

Find the domain of the rational function.
$$f ( x ) = \frac { 2 } { x - 2 }$$

A) domain of $f : ( \infty , \infty )$
B) domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$
C) domain of $f : ( \infty , - 2 ) \cup ( - 2 , \infty )$
D) domain of $f : ( \infty , 2 ) \cup ( 2 , \infty )$

Answer 3

D

Question 4

Find the domain of the rational function.
$$f ( x ) = \frac { x ^ { 2 } - 9 } { x ^ { 2 } - 8 x + 12 }$$

A) domain of $f : ( \infty , - 3 ) \cup ( - 3,3 ) \cup ( 3 , \infty )$
B) domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$
C) domain of $f : ( \infty , 2 ) \cup ( 2,6 ) \cup ( 6 , \infty )$
D) domain of $f : ( \infty , - 6 ) \cup ( - 6 , - 2 ) \cup ( - 2 , \infty )$

Question 5

Find the domain of the rational function.
$$f ( x ) = \frac { x - 3 } { 5 - x }$$

A) domain of $\mathrm { f } : ( \infty , 5 ) \cup ( 5 , \infty )$
B) domain of $f : ( \infty , \infty )$
C) domain of $f : ( \infty , - 5 ) \cup ( - 5 , \infty )$
D) domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$

Question 6

Find the function value.
$$f ( x ) = \frac { x - 3 } { 2 x } ; f ( - 3 )$$

A) $\frac { 1 } { 2 }$
B) 2
C) 1
D) $- 3$

Question 7

Solve the problem.
Suppose a cost-benefit model is given by $y = \frac { 5.3 x } { 100 - x }$ , where $y$ is the cost in thousands of dollars for removing $x$ percent of a given pollutant. Find the cost for removing $60 \%$ of the pollutant.

A) $\$ 79.5$ thousands
B) $\$ 9.2$ thousands
C) $\$ 8$ thousands
D) $\$ 2$ thousands

Question 8

Find the domain of the rational function.
$$f ( x ) = \frac { x - 3 } { 6 }$$

A) domain of $\mathrm { f } : ( \infty , \infty )$
B) domain of $f : ( \infty , - 3 ) \cup ( - 3 , \infty )$
C) domain of $f : ( \infty , 3 ) \cup ( 3 , \infty )$
D) domain of $\mathrm { f } : ( \infty , 0 ) \cup ( 0 , \infty )$

Question 9

Solve the problem.
A drug is injected into a patient and the concentration of the drug is monitored. The drug's concentration, $C ( t )$ , in milligrams after $t$ hours is modeled by $C ( t ) = \frac { 6 t } { 3 t ^ { 2 } + 2 }$ . Estimate the drug's concentration after 4 hours. (Round to the nearest hundredth.)

A) $0.48 \mathrm { mg }$
B) $1.71 \mathrm { mg }$
C) $0.35 \mathrm { mg }$
D) $1.58 \mathrm { mg }$

Question 10

Solve the problem.
To calculate the drug dosage for a child, a pharmacist may use the formula $\mathrm { d } ( \mathrm { x } ) = \frac { \mathrm { Dx } } { \mathrm { x } + 10 } , 0 \leq \mathrm { x } \leq 12$ . The child's age is $x$ and the adult dosage is D. What is the dosage for an 3 -year old child if the adult dosage is 40 mg? (Round to the nearest tenth.)

A) $3.1 \mathrm { mg }$
B) $12.0 \mathrm { mg }$
C) $9.2 \mathrm { mg }$
D) $40.0 \mathrm { mg }$

Question 11

Find the domain of the rational function.
$$f ( x ) = \frac { x ^ { 2 } - 9 } { x ^ { 2 } + 9 x + 14 }$$

A) domain of $f : ( \infty , - 3 ) \cup ( - 3,3 ) \cup ( 3 , \infty )$
B) domain of $f : ( \infty , 2 ) \cup ( 2,7 ) \cup ( 7 , \infty )$
C) domain of $f : ( \infty , - 7 ) \cup ( - 7 , - 2 ) \cup ( - 2 , \infty )$
D) domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$

Question 12

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.
Find all horizontal asymptotes of the graph.

A) x = 4, x = -4
B) x = -6
C) y = -6
D) y = 4, y = -4

Question 13

Find the function value.
$$f ( x ) = \frac { x ^ { 3 } - 2 } { x ^ { 2 } - 2 } ; f ( 2 )$$

A) $\frac { 3 } { 2 }$
B) 4
C) 1
D) 3

Question 14

Find the function value.
$$f ( x ) = \frac { x ^ { 2 } + 4 } { x ^ { 3 } - 7 x } ; f ( - 5 )$$

A) $- \frac { 5 } { 18 }$
B) $- \frac { 29 } { 90 }$
C) $- \frac { 29 } { 132 }$
D) $- \frac { 29 } { 125 }$

Question 15

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.
Find f(-2).

A) 1
B) 0
C) -2
D) 2

Question 16

Find the domain of the rational function.
$$f ( x ) = \frac { x ^ { 2 } - 81 } { x ^ { 2 } + 7 x - 18 }$$

A) domain of $f : ( \infty , - 9 ) \cup ( - 9,9 ) \cup ( 9 , \infty )$
B) domain of $f : ( \infty , - 9 ) \cup ( - 9,2 ) \cup ( 2 , \infty )$
C) domain of $f : ( \infty , - 2 ) \cup ( - 2,9 ) \cup ( 9 , \infty )$
D) domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$

Question 17

Find the function value.
$$f ( x ) = \frac { x + 6 } { 14 x - 13 } ; f ( - 2 )$$

A) $\frac { 4 } { 41 }$
B) $\frac { 4 } { 71 }$
C) $- \frac { 4 } { 15 }$
D) $- \frac { 4 } { 41 }$

Question 18

Find the domain of the rational function.
$$f ( x ) = \frac { 4 } { x + 4 }$$

A) domain of $f : ( \infty , - 4 ) \cup ( - 4 , \infty )$
B) domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$
C) domain of $\mathrm { f } : ( \infty , 4 ) \cup ( 4 , \infty )$
D) domain of $\mathrm { f } : ( \infty , \infty )$

Question 19

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.
Find all vertical asymptotes of the graph.

A) x = 4, x = -4
B) x = -6
C) y = 4, y = -4
D) y = -6

Question 20

Solve the problem.
A company that produces radios has costs given by the function C(x) = 25x + 25,000, where x is the number of radios manufactured and C(x) is measured in dollars. The average cost to manufacture each radio is given by $\bar { C } ( x ) = \frac { 25 x + 25,000 } { x }$ . Find $\bar { C } ( 50 )$ . (Round to the nearest dollar, if necessary.)

A) $\$ 75$
B) $\$ 525$
C) $\$ 74$
D) $\$ 535$

Question 21

Simplify the rational expression. If the rational expression cannot be simplified, so state.
$$\frac { y ^ { 3 } - 64 } { y - 4 }$$

A) $y ^ { 2 } - 16$
B) $\frac { 1 } { y - 4 }$
C) $y ^ { 2 } + 4 y + 16$
D) Cannot be simplified

Question 22

Simplify the rational expression. If the rational expression cannot be simplified, so state.
$\frac { 15 x ^ { 2 } + 24 x + 9 } { 3 x + 3 }$

A) $\frac { 5 x + 3 } { 3 x }$
B) $\frac { 5 x + 24 } { 3 x + 5 }$
C) $5 x + 3$
D) Cannot be simplified

Question 23

$$\text { The rational function } f ( x ) = \frac { 400 + 5 x } { x } \text { models the cost, } f ( x ) \text {, in dollars, to produce } x \text { bobble-head figures. The graph is show. }$$
According to the cost model, is it possible to remove 100% of the trash from American highways?

A) Yes
B) No

Question 24

Simplify the rational expression. If the rational expression cannot be simplified, so state.
$$\frac { 3 x + 2 } { 12 x ^ { 2 } + 23 x + 10 }$$

A) Cannot be simplified
B) $\frac { 3 x + 4 } { 4 x + 23 }$
C) $\frac { 3 x } { 4 x + 5 }$
D) $\frac { 1 } { 4 x + 5 }$

Question 25

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.
Is 4 a function value of f?

A) Yes
B) No

Question 26

Multiply as indicated.
$\frac { 2 x ^ { 2 } } { 4 } \cdot \frac { 8 } { x ^ { 3 } }$

A) $\frac { 4 x ^ { 2 } } { x ^ { 3 } }$
B) $\frac { x } { 4 }$
C) $\frac { 16 x ^ { 2 } } { 4 x ^ { 3 } }$
D) $\frac { 4 } { x }$

Question 27

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.
How does the graph indicate that f(-4) does not exist?

A) There is a horizontal asymptote at x = -4 indicated by a dashed line.
B) There is a horizontal asymptote at y = -4 indicated by a dashed line.
C) There is a vertical asymptote at y = -4 indicated by a dashed line.
D) There is a vertical asymptote at x = -4 indicated by a dashed line.

Question 28

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.
Is -3 a function value of f?

A) Yes
B) No

Question 29

Simplify the rational expression. If the rational expression cannot be simplified, so state.
$$\frac { y ^ { 2 } + 6 y + 9 } { y ^ { 2 } + 9 y + 18 }$$

A) $\frac { 6 y + 1 } { 9 y + 2 }$
B) $\frac { 6 y + 9 } { 9 y + 18 }$
C) $\frac { y + 3 } { y + 6 }$
D) Cannot be simplified

Question 30

Simplify the rational expression. If the rational expression cannot be simplified, so state.
$$\frac { 2 x ^ { 2 } - 18 x + 28 } { x - 7 }$$

A) Cannot be simplified
B) $\frac { 1 } { x - 7 }$
C) $2 x ^ { 2 } - 22$
D) $2 x - 4$

Question 31

Multiply as indicated.
$$\frac { x ^ { 2 } - 7 x + 10 } { x ^ { 2 } - 21 x + 110 } \cdot \frac { x ^ { 2 } - 15 x + 44 } { x ^ { 2 } - 13 x + 40 }$$

A) $\frac { \left( x ^ { 2 } - 7 x + 10 \right) \left( x ^ { 2 } - 15 x + 44 \right) } { \left( x ^ { 2 } - 21 x + 110 \right) \left( x ^ { 2 } - 13 x + 40 \right) }$
B) $\frac { ( x - 2 ) } { ( x - 8 ) }$
C) $\frac { ( x + 2 ) ( x + 4 ) } { ( x + 10 ) ( x + 8 ) }$
D) $\frac { ( x - 2 ) ( x - 4 ) } { ( x - 10 ) ( x - 8 ) }$

Question 32

Simplify the rational expression. If the rational expression cannot be simplified, so state.
$$\frac { 4 x + 20 } { 5 x ^ { 2 } + 31 x + 30 }$$

A) $\frac { 4 x + 5 } { 5 x + 31 }$
B) $\frac { 4 x } { 5 x + 6 }$
C) $\frac { 4 } { 5 x + 6 }$
D) Cannot be simplified

Question 33

Use the equation to solve the problem.
$$\text { Kurberof thures pasetsed }$$
What is the horizontal asymptote of the graph? What does this mean about the cost to produce x bobble-heads?

A) y = 5; As the number of bobble-head figures produced increases, the cost is approaching $5.
B) y = 400; As the number of bobble-head figures produced increases, the cost is approaching $400.
C) There is no horizontal asymptote.
D) y = 0; As the number of bobble-head figures produced increases, the cost is approaching $0

Question 34

Simplify the rational expression. If the rational expression cannot be simplified, so state.
$$\frac { y ^ { 2 } + 5 y - 24 } { y ^ { 2 } + 3 y - 40 }$$

A) $\frac { 5 y - 3 } { 3 y - 5 }$
B) Cannot be simplified
C) $\frac { 5 y - 24 } { 3 y - 40 }$
D) $\frac { y - 3 } { y - 5 }$

Question 35

$$\text { The rational function } f ( x ) = \frac { 120 x } { 100 - x } \text { models the cost, } f ( x ) \text {, in millions of dollars, to remove } x \% \text { of the trash from American }$$ highways. The graph is shown. Use the equation to solve the problem.

What value of x must be excluded from the rational function's domain?

A) 10
B) 100
C) 1000
D) 1

Question 36

Use the equation to solve the problem.
$$\text { Kurberof thures pasetsed }$$
Find and interpret f(40).

A) 15; It costs $15 to produce 40 bobble-head figures.
B) 405; It costs $405 to produce 40 bobble-head figures.
C) 5; It costs $5 to produce 40 bobble-head figures.
D) 600; It costs $600 to produce 40 bobble-head figures.

Question 37

Multiply as indicated.
$$\frac { 5 y } { 10 y + 5 } \cdot \frac { 4 y + 2 } { 7 }$$

A) $\frac { y } { 7 }$
B) $\frac { 2 y } { 7 }$
C) $\frac { 2 y } { 35 }$
D) $\frac { 2 } { 7 }$

Question 38

The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.
How can you tell that this is not the graph of a polynomial function?

A) The graph is not continuous.
B) The value of f(1) is not equal to 1.
C) The graph is continuous.
D) The graph is not a parabola.

Question 39

$$\text { The rational function } f ( x ) = \frac { 120 x } { 100 - x } \text { models the cost, } f ( x ) \text {, in millions of dollars, to remove } x \% \text { of the trash from American }$$ highways. The graph is shown. Use the equation to solve the problem.

Find and interpret f(80).

A) 48; The cost to remove 80% of the trash is $48 million.
B) 48; The cost to remove 20% of the trash is $48 million.
C) 48; The cost to remove 80% of the trash is $48 .
D) 20; The cost to remove 80% of the trash is $20 million.

Question 40

Simplify the rational expression. If the rational expression cannot be simplified, so state.
$$\frac { 7 x ^ { 2 } + 21 x ^ { 3 } } { 11 x + 33 x ^ { 2 } }$$

A) $\frac { 7 } { 11 }$
B) $\frac { 7 + 21 x ^ { 3 } } { 11 x + 33 }$
C) $\frac { 7 x } { 11 }$
D) Cannot be simplified

Question 41

Multiply as indicated.
$$\frac { x ^ { 2 } + 7 x + 12 } { x ^ { 2 } + 9 x + 20 } \cdot \frac { x ^ { 2 } + 5 x } { x ^ { 2 } - 3 x - 18 }$$

A) $\frac { x } { x ^ { 2 } + 9 x + 20 }$
B) $\frac { x } { x - 6 }$
C) $\frac { x ^ { 2 } + 5 x } { x - 6 }$
D) $\frac { 1 } { x - 6 }$

Question 42

Divide as indicated.
$$\frac { x ^ { 2 } - 7 x + x y - 7 y } { 9 x ^ { 2 } - 9 y ^ { 2 } } \div \frac { x - 7 } { 10 x - 10 y }$$

A) 1
B) $\frac { 10 \left( x ^ { 2 } - 7 x + x y - 7 y \right) } { 9 ( x + y ) ( x - 7 ) }$
C) $\frac { 10 } { 9 }$
D) $\frac { ( x - 7 ) ^ { 2 } } { 90 ( x - y ) ^ { 2 } }$

Question 43

Divide as indicated.
$$\frac { x ^ { 2 } - 17 x + 66 } { 11 - x } \div ( x + 6 )$$

A) $- \frac { x - 6 } { x + 6 }$
B) $- 1$
C) $- ( x - 6 ) ( x + 6 )$
D) $- \frac { x + 6 } { x - 6 }$

Question 44

Multiply as indicated.
$$\frac { x ^ { 3 } + 1 } { x ^ { 3 } - x ^ { 2 } + x } \cdot \frac { 5 x } { - 50 x - 50 }$$

A) $- \frac { x ^ { 2 } + 1 } { 10 }$
$B ) = \frac { x ^ { 3 } + 1 } { 10 ( x + 1 ) }$
C) $- \frac { 1 } { 10 }$
D) $\frac { x + 1 } { 10 ( - x - 1 ) }$

Question 45

Divide as indicated.
$$\frac { x ^ { 2 } - 20 x + 100 } { 6 x - 60 } \div \frac { 3 x - 30 } { 18 }$$

A) $\frac { x ^ { 2 } - 20 x + 100 } { ( x - 10 ) ^ { 2 } }$
B) 1
C) 18
D) $\frac { ( x - 10 ) ^ { 2 } } { 36 }$

Question 46

Divide as indicated.
$$\frac { x ^ { 2 } + 16 x + 64 } { x ^ { 2 } + 17 x + 72 } \div \frac { x ^ { 2 } + 8 x } { x ^ { 2 } + 3 x - 54 }$$

A) $\frac { x } { x ^ { 2 } + 17 x + 72 }$
B) $\frac { x - 6 } { x ^ { 2 } + 9 x }$
C) $\frac { x - 6 } { x }$
D) $x - 6$

Question 47

Add. Simplify the result, if possible.
$$\frac { x } { x + 5 } + \frac { - 8 } { x + 5 }$$

A) $- \frac { 8 } { 5 }$
B) $\frac { - 8 x } { x ^ { 2 } + 10 x + 25 }$
C) $\frac { x - 8 } { x + 5 }$
D) $\frac { - 7 x } { x + 5 }$

Question 48

Add. Simplify the result, if possible.
$$\frac { 7 } { 8 x ^ { 2 } } + \frac { 5 } { 8 x ^ { 2 } }$$

A) 3
B) $\frac { 3 } { 4 x ^ { 4 } }$
C) $\frac { 3 } { 2 x ^ { 2 } }$
D) $\frac { 2 } { 3 x ^ { 2 } }$

Question 49

Divide as indicated.
$$\frac { ( y - 5 ) ^ { 2 } } { 3 } \div \frac { 3 y - 15 } { 9 }$$

A) $y - 5$
B) $\frac { 3 ( y - 5 ) ^ { 2 } } { 3 y - 15 }$
C) $\frac { ( y - 5 ) ^ { 3 } } { 9 }$
D) $\frac { 1 } { y - 5 }$

Question 50

Multiply as indicated.
$\frac { 2 x ^ { 3 } } { 3 } \cdot \frac { 9 } { x ^ { 2 } }$

A) $\frac { 6 } { x }$
B) $6 x$
C) $\frac { x } { 6 }$
D) $\frac { 6 x ^ { 2 } } { x ^ { 3 } }$

Question 51

Multiply as indicated.
$$\frac { 6 x - 6 } { x } \cdot \frac { 9 x ^ { 2 } } { 8 x - 8 }$$

A) $\frac { 48 x ^ { 2 } + 96 x + 48 } { 9 x ^ { 3 } }$
B) $\frac { 27 x } { 4 }$
C) $\frac { 54 x ^ { 3 } - 54 x ^ { 2 } } { 8 x ^ { 2 } - 8 x }$
D) $\frac { 4 } { 27 x }$

Question 52

Add. Simplify the result, if possible.
$\frac { 4 } { 9 x ^ { 2 } } + \frac { 2 } { 9 x ^ { 2 } }$

A) $\frac { 2 } { 3 x ^ { 4 } }$
B) 2
C) $\frac { 2 } { 3 x ^ { 2 } }$
D) $\frac { 3 } { 2 x ^ { 2 } }$

Question 53

Divide as indicated.
$$\frac { 3 x - 3 } { x } \div \frac { 4 x - 4 } { 2 x ^ { 2 } }$$

A) $\frac { 6 x ^ { 3 } - 6 x ^ { 2 } } { 4 x ^ { 2 } - 4 x }$
B) $\frac { 3 x } { 2 }$
C) $\frac { 2 } { 3 x }$
D) $\frac { 12 x ^ { 2 } + 24 x + 12 } { 2 x ^ { 3 } }$

Question 54

Divide as indicated.
$\frac { 4 x ^ { 2 } } { 5 } \div \frac { x ^ { 3 } } { 30 }$

A) $\frac { 24 x ^ { 2 } } { x ^ { 3 } }$
B) $\frac { x } { 24 }$
C) $\frac { 24 } { x }$
D) $\frac { 120 x ^ { 2 } } { 5 x ^ { 3 } }$

Question 55

Multiply as indicated.
$$\frac { x ^ { 2 } + 6 x + 8 } { x ^ { 2 } + 8 x + 12 } \cdot \frac { x ^ { 2 } + 6 x } { x ^ { 2 } + 11 x + 28 }$$

A) $\frac { x } { x + 7 }$
B) $\frac { x ^ { 2 } + 6 x } { x + 7 }$
C) $\frac { x } { x ^ { 2 } + 8 x + 12 }$
D) $\frac { 1 } { x + 7 }$

Question 56

Divide as indicated.
$$\frac { 21 x - 21 } { 11 } \div \frac { 7 x - 7 } { 66 }$$

A) $\frac { 147 ( x - 1 ) ^ { 2 } } { 726 }$
B) 18
C) $\frac { 6 ( 21 x - 21 ) } { 7 x - 7 }$
D) $\frac { 1 } { 18 }$

Question 57

Divide as indicated.
$$\frac { x ^ { 2 } + 15 x + 54 } { x ^ { 2 } + 16 x + 63 } \div \frac { x ^ { 2 } + 6 x } { x ^ { 2 } + 17 x + 70 }$$

A) $\frac { x + 10 } { x }$
B) $\frac { x + 10 } { x ^ { 2 } + 7 x }$
C) $x + 10$
D) $\frac { x } { x ^ { 2 } + 16 x + 63 }$

Question 58

Multiply as indicated.
$2 x ^ { 4 } \cdot \frac { x ^ { 2 } y } { 6 x ^ { 3 } y ^ { 2 } }$

A) $\frac { x ^ { 3 } } { 3 y }$
B) $\frac { x ^ { 5 } } { 3 y }$
C) $\frac { 1 } { 6 x y }$
D) $\frac { x ^ { 6 } y } { 3 x ^ { 3 } y ^ { 2 } }$

Question 59

Multiply as indicated.
$$\frac { x ^ { 2 } + 11 x + 28 } { x ^ { 2 } + 13 x + 42 } \cdot \frac { x ^ { 2 } + 11 x + 30 } { x ^ { 2 } + 9 x + 20 }$$

A) $\frac { 1 } { x + 5 }$
B) $\frac { x + 4 } { x + 6 }$
C) $\frac { x + 6 } { x + 5 }$
D) 1

Question 60

Add. Simplify the result, if possible.
$\frac { 4 } { 13 x } + \frac { 6 } { 13 x }$
B) 1
C) $\frac { 10 } { 26 x }$
D) $\frac { 10 } { 13 x }$

A) $\frac { 13 x } { 10 }$
$26 x$

Question 61

Subtract. Simplify the result, if possible.
$\frac { 19 } { 22 x } - \frac { 6 } { 22 x }$

A) $\frac { 13 } { 44 x }$
B) 13
C) $\frac { 22 x } { 13 }$
D) $\frac { 13 } { 22 x }$

Question 62

Subtract. Simplify the result, if possible.
$\frac { 15 } { 8 x ^ { 2 } } - \frac { 12 } { 8 x ^ { 2 } }$

A) 3
B) $\frac { 3 } { 8 x ^ { 2 } }$
C) $\frac { 3 } { 8 x ^ { 4 } }$
D) $\frac { 8 } { 3 x ^ { 2 } }$

Question 63

Perform the indicated operations. Simplify the result, if possible.
$$\frac { 4 } { x } + \frac { 8 } { x - 2 }$$

A) $\frac { 12 x - 8 } { x ( x - 2 ) }$
B) $\frac { 12 x - 8 } { x ( 2 - x ) }$
C) $\frac { 8 x - 12 } { x ( 2 - x ) }$
D) $\frac { 8 x - 12 } { x ( x - 2 ) }$

Question 64

Find the least common denominator of the rational expressions.
$\frac { 1 } { 40 x } , \frac { 1 } { 5 x ^ { 2 } }$ , and $\frac { 1 } { 8 x ^ { 3 } }$

A) $40 x ^ { 2 }$
B) $40 x ^ { 5 }$
C) $8 x ^ { 3 }$
D) $40 x ^ { 3 }$

Question 65

Perform the indicated operations. Simplify the result, if possible.
$$\frac { x + 7 } { x ^ { 2 } - 3 x - 10 } + \frac { 3 x + 8 } { x ^ { 2 } - 11 x + 30 }$$

A) $4 x + 15$
B) $\frac { 4 x ^ { 2 } + 15 x - 26 } { ( x - 5 ) ( x + 2 ) ( x - 6 ) }$
C) $\frac { 4 x ^ { 2 } + 15 x - 26 } { ( x + 5 ) ( x - 2 ) ( x + 6 ) }$
D) $\frac { 4 x + 15 } { 2 x ^ { 2 } - 14 x + 20 }$

Question 66

Subtract. Simplify the result, if possible.
$\frac { 2 x ^ { 2 } } { x - 1 } - \frac { 2 x } { x - 1 }$

A) $2 x$
B) $\frac { 2 x } { x - 1 }$
C) 0
D) $\frac { 2 x ( x + 1 ) } { x - 1 }$

Question 67

Find the least common denominator of the rational expressions.
$\frac { 15 } { x ^ { 2 } + 2 x - 15 }$ and $\frac { 4 x - 20 } { - 4 x - 20 }$

A) $- 4 ( x - 3 ) ( x - 5 )$
B) $- 4 ( x - 3 ) ( x + 5 )$
C) $- 4 ( x + 3 ) ( x - 5 )$
D) $- 4 ( x + 3 ) ( x + 5 )$

Question 68

Find the least common denominator of the rational expressions.
$\frac { 5 } { y ^ { 2 } - 4 } , \frac { 2 y } { y ^ { 2 } + 4 y + 4 }$ , and $\frac { 5 y } { 2 y ^ { 2 } + 5 y + 2 }$

A) $( y - 2 ) ( y + 2 ) ( 2 y + 1 )$
B) $( y - 2 ) ( y - 2 ) ( y + 2 ) ( 2 y + 1 )$
C) $( y - 2 ) ( y + 2 ) ( y + 2 ) ( 2 y + 1 )$
D) $( y - 2 ) ( y + 2 )$

Question 69

Find the least common denominator of the rational expressions.
$\frac { x + 5 } { x ^ { 2 } - 5 x - 6 }$ and $\frac { x + 9 } { x ^ { 2 } - 9 x + 18 }$

A) $( x + 1 ) ( x - 6 )$
B) $( x - 6 ) ( x - 3 )$
C) $( x - 1 ) ( x + 6 ) ( x - 3 )$
D) $( x + 1 ) ( x - 6 ) ( x - 3 )$

Question 70

Perform the indicated operations. Simplify the result, if possible.
$$\frac { 3 } { y ^ { 2 } - 3 y + 2 } + \frac { 7 } { y ^ { 2 } - 1 }$$

A) $\frac { 42 y - 11 } { ( y - 1 ) ( y + 1 ) ( y - 2 ) }$
B) $\frac { 10 y - 11 } { ( y - 1 ) ( y + 1 ) ( y - 2 ) }$
C) $\frac { 11 y - 10 } { ( y - 1 ) ( y + 1 ) ( y - 2 ) }$
D) $\frac { 10 y - 11 } { ( y - 1 ) ( y - 2 ) }$

Question 71

Find the least common denominator of the rational expressions.
$\frac { 1 } { - 3 x }$ and $\frac { 2 } { x ^ { 2 } + 7 x }$

A) $- 3 x ( x + 7 )$
B) $- 3 x + 7$
C) $- 3 x ^ { 2 } + 2$
D) $- 3 x ^ { 2 } + 7$

Question 72

Subtract. Simplify the result, if possible.
$$\frac { x } { x ^ { 2 } - 16 } - \frac { 7 } { x ^ { 2 } + 5 x + 4 }$$

A) $\frac { x ^ { 2 } - 6 x + 28 } { ( x - 4 ) ( x + 4 ) }$
B) $\frac { x ^ { 2 } - 6 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$
C) $\frac { x ^ { 2 } + 6 x + 28 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$
D) $\frac { x ^ { 2 } - 6 x + 28 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$

Question 73

Find the least common denominator of the rational expressions.
$\frac { x - 1 } { x ^ { 2 } + 9 x + 8 }$ and $\frac { 1 } { x ^ { 2 } + x }$

A) $x + 1$
B) $( x + 1 ) ^ { 2 }$
C) $x ( x + 1 ) ^ { 2 }$
D) $x ( x + 8 ) ( x + 1 )$

Question 74

Perform the indicated operations. Simplify the result, if possible.
$$\frac { x } { x ^ { 2 } - 16 } - \frac { 4 } { x ^ { 2 } + 5 x + 4 }$$

A) $\frac { x ^ { 2 } - 3 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$
B) $\frac { x ^ { 2 } - 3 x + 16 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$
C) $\frac { x ^ { 2 } - 3 x + 16 } { ( x - 4 ) ( x + 4 ) }$
D) $\frac { x ^ { 2 } + 3 x + 16 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$

Question 75

Find the least common denominator of the rational expressions.
$\frac { 3 } { x }$ and $\frac { 3 } { x + 6 }$

A) $- 6$
B) 6
C) $x ( x + 6 )$
D) $x + 6$

Question 76

Add. Simplify the result, if possible.
$$\frac { 3 x + 5 } { x ^ { 2 } + 7 x + 10 } + \frac { - 3 - 2 x } { x ^ { 2 } + 7 x + 10 }$$

A) $\frac { 1 } { x ^ { 2 } + 7 x + 10 }$
B) $\frac { 1 } { x + 5 }$
C) $\frac { x - 2 } { x ^ { 2 } + 7 x + 10 }$
D) $\frac { 1 } { x + 2 }$

Question 77

Find the least common denominator of the rational expressions.
$\frac { x + 5 } { x ^ { 2 } - 5 x }$ and $\frac { - 10 } { x ^ { 2 } - 7 x + 10 }$

A) $x ( x - 5 ) ( x - 2 )$
B) $( x - 3 ) ^ { 2 }$
C) $x ( x - 3 ) ( x - 2 )$
D) $x ( x - 3 ) ^ { 2 }$

Question 78

Perform the indicated operations. Simplify the result, if possible.
$$\frac { 3 } { x } + \frac { 3 } { x - 4 }$$

A) $\frac { 12 x - 6 } { x ( 4 - x ) }$
B) $\frac { 6 x - 12 } { x ( x - 4 ) }$
C) $\frac { 6 x - 12 } { x ( 4 - x ) }$
D) $\frac { 12 x - 6 } { x ( x - 4 ) }$

Question 79

Perform the indicated operations. Simplify the result, if possible.
$\frac { 2 } { x ^ { 2 } } - \frac { 8 } { x }$

A) $\frac { 8 x - 2 } { x }$
B) $\frac { 2 - 8 x } { x ^ { 2 } }$
C) $\frac { 2 + 8 x } { x ^ { 2 } }$
D) $\frac { 2 x + 8 } { x ^ { 2 } }$

Question 80

Add. Simplify the result, if possible.
$$\frac { x ^ { 2 } - 5 x } { x - 2 } + \frac { 6 } { x - 2 }$$

A) $x + 3$
B) $\frac { x ^ { 2 } - 5 x + 6 } { 2 }$
C) $x - 3$
D) $x - 2$