Deck 6: Rational Expressions

Question 1

Find the domain of the rational function.
$f ( x ) = \frac { x ^ { 3 } + 6 x ^ { 4 } } { x ^ { 2 } + 64 }$

A) $\left\{ x \mid x \text { is a real number and } x \neq 0 , x \neq - \frac { 1 } { 6 } \right\}$
B) $\{ x \mid x \text { is a real number and } x \neq - 8 \}$
C) $\{ x \mid x \text { is a real number and } x \neq - 64 \}$
D) $\{ x \mid x \text { is a real number } \}$

Answer 1

D

Question 2

Find the domain of the rational function.
$f ( x ) = \frac { 6 x - 5 } { 7 }$

A) $\{ x \mid x \text { is a real number and } x \neq 7 \}$
B) $\{ x \mid x \text { is a real number and } x \neq 0 \}$
C) $\left\{ x \mid x \text { is a real number and } x \neq \frac { 5 } { 6 } \right\}$
D) $\{ x \mid x \text { is a real number } \}$

Answer 2

D

Question 3

Find the domain of the rational function. $f ( x ) = \frac { 9 x ^ { 2 } - 4 } { 5 x - 30 }$

A) $\left\{ x \mid x \text { is a real number and } x \neq \frac { 2 } { 3 } , x \neq - \frac { 2 } { 3 } \right\}$
B) $\{ x \mid x \text { is a real number and } x \neq 30 \}$
C) $\{ x \mid x \text { is a real number and } x \neq 6 \}$
D) $\{ x \mid x \text { is a real number }$

Answer 3

C

Question 4

Find the domain of the rational function.
$f ( x ) = \frac { 1 - 6 x } { x ^ { 3 } - 18 x ^ { 2 } + 45 x }$

A) $\{ x \mid x \text { is a real number and } x \neq - 15 , x \neq 3 , x \neq 0 \}$
B) $\{ x \mid x \text { is a real number and } x \neq 15 , x \neq 3 , x \neq 0 \}$
C) $\left\{ x \mid x \text { is a real number and } x \neq 15 , x \neq 3 , x \neq \frac { 1 } { 6 } , x \neq 0 \right\}$
D) $\left\{ x \mid x \text { is a real number and } x \neq 15 , x \neq 3 , x \neq \frac { 1 } { 6 } \right\}$

Question 5

Simplify the rational expression.
$\frac { 12 x - 30 x ^ { 2 } } { 6 x }$

A) 1
B) 0
C) $2 + 5 x$
D) $2 - 5 x$

Question 6

Simplify the rational expression.
$$\frac { 12 - 4 x } { 21 - 7 x }$$

A) 3
B) $\frac { 4 } { 7 }$
C) $- 3$
D) $- \frac { 4 } { 7 }$

Question 7

Find the domain of the rational function.
$f ( x ) = \frac { x ^ { 2 } - 4 } { x ^ { 2 } + 15 x + 56 }$

A) $\{ x \mid x \text { is a real number and } x \neq 0 \}$
B) $\{ x \mid x \text { is a real number and } x \neq 8 , x \neq - 7 \}$
C) $\{ x \mid x \text { is a real number and } x \neq 2 , x \neq - 2 \}$
D) $\{ x \mid x \text { is a real number and } x z - 8 , x \neq - 7 \}$

Question 8

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

A) $\{ x \mid x \text { is a real number and } x \neq 0 \}$
B) $\{ x \mid x \text { is a real number and } x \neq 8 \}$
C) $\{ x \mid x \text { is a real number and } x \neq - 8 \}$
D) $\{ x \mid x \text { is a real number and } x \neq 0 , x \neq 8 \}$

Question 9

$f ( x ) = \frac { 1 - 6 x } { x ^ { 3 } - 18 x ^ { 2 } + 45 x }$

A) $\{ x \mid x \text { is a real number and } x \neq - 15 , x \neq 3 , x \neq 0 \}$
B) $\{ x \mid x \text { is a real number and } x \neq 15 , x \neq 3 , x \neq 0 \}$
C) $\left\{ x \mid x \text { is a real number and } x \neq 15 , x \neq 3 , x \neq \frac { 1 } { 6 } , x \neq 0 \right\}$
D) $\left\{ x \mid x \text { is a real number and } x \neq 15 , x \neq 3 , x \neq \frac { 1 } { 6 } \right\}$

Question 10

Find the domain of the rational function.
$f ( x ) = \frac { x } { 7 x - 8 }$

A) $\left\{ x \mid x \text { is a real number and } x \neq - \frac { 7 } { 8 } , x \neq 0 \right\}$
B) $\left\{ x \mid x \text { is a real number and } x \neq \frac { 7 } { 8 } , x \neq 0 \right\}$
C) $\left\{ x \mid x \text { is a real number and } x \neq \frac { 8 } { 7 } \right\}$
D) $\left\{ x \mid x \text { is a real number and } x \neq - \frac { 8 } { 7 } \right\}$

Question 11

Simplify the rational expression.
$$\frac { 5 x - 20 } { 4 - x }$$

A) $\frac { 1 } { 5 }$
B) $- 5$
C) 5
D) $- \frac { 1 } { 5 }$

Question 12

Find the domain of the rational function.
$f ( x ) = \frac { 8 x } { 5 - x }$

A) $\{ x \mid x \text { is a real number and } x \neq - 5 \}$
B) $\{ x \mid x \text { is a real number and } x \neq 0 , x \neq - 5 \}$
C) $\{ x \mid x \text { is a real number and } x \neq 0 \}$
D) $\{ x \mid x \text { is a real number and } x \neq 5 \}$

Question 13

Simplify the rational expression.
$\frac { 3 - x } { x - 3 }$

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

Question 14

Find the domain of the rational function.
$f ( r ) = \frac { r ^ { 2 } + 24 r } { 6 r }$

A) $\left\{ \mathrm { r } \mid \mathrm { r } \text { is a real number and } \mathrm { r } \neq \frac { 1 } { 6 } \right\}$
B) $\{ r \mid r \text { is a real number and } r \neq 0 \text { and } r \neq - 24 \}$
C) $\{ \mathrm { r } \mid \mathrm { r } \text { is a real number and } \mathrm { r } \neq 6 \}$
D) $\{ \mathrm { r } \mid \mathrm { r } \text { is a real number and } \mathrm { r } \neq 0 \}$

Question 15

Find the domain of the rational function.
$f ( x ) = \frac { x ^ { 2 } - 64 } { x ^ { 2 } - 7 x + 10 }$

A) $\{ x \mid x \text { is a real number and } x \neq - 2 , x \neq - 5 \}$
B) $\{ x \mid x \text { is a real number and } x \neq 2 , x \neq 5 \}$
C) $\{ x \mid x \text { is a real number and } x \neq 8 , x \neq - 8 \}$
D) $\{ x \mid x \text { is a real number and } x \neq 0 \}$

Question 16

Simplify the rational expression.
$$\frac { 3 x - 12 } { 8 - 2 x }$$

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

Question 17

Simplify the rational expression.
$$\frac { 2 - x } { 5 x - 10 }$$

A) $- \frac { 1 } { 5 }$
B) $- 5$
C) 5
D) $\frac { 1 } { 5 }$

Question 18

Find the domain of the rational function.
$f ( x ) = \frac { 1 - 2 x } { x ^ { 2 } - 20 x + 96 }$

A) $\{ x \mid x \text { is a real number and } x \neq - 12 , x \neq 8 \}$
B) $\left\{ x \mid x \text { is a real number and } x \neq 12 , x \neq 8 , x \neq \frac { 1 } { 2 } , x \neq 0 \right\}$
C) $\{ x \mid x \text { is a real number and } x \neq 12 , x \neq 8 \}$
D) $\left\{ x \mid x \text { is a real number and } x \neq 12 , x \neq 8 , x \neq \frac { 1 } { 2 } \right\}$

Question 19

Find the domain of the rational function.
$f ( x ) = \frac { - 7 } { 5 x + 4 }$

A) $\{ x \mid x \text { is a real number and } x \neq 0 \}$
B) $\left\{ x \mid x \text { is a real number and } x \neq \frac { 4 } { 5 } \right\}$
C) $\left\{ x \mid x \text { is a real number and } x z - \frac { 4 } { 5 } \right\}$
D) $\{ x \mid x \text { is a real number and } x \neq 7 \}$

Question 20

Simplify the rational expression.
$$\frac { 3 x - 6 } { 2 x - 4 }$$

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

Question 21

Simplify the rational expression.
$\frac { 25 x ^ { 2 } - 15 x + 9 } { 875 x ^ { 3 } + 189 }$

A) $\frac { 1 } { 7 ( 5 x - 3 ) }$
B) $\frac { 7 } { 5 x - 3 }$
C) $\frac { 7 } { 5 x + 3 }$
D) $\frac { 1 } { 7 ( 5 x + 3 ) }$

Question 22

Simplify the rational expression.
$$\frac { x ^ { 3 } - 216 } { x - 6 }$$

A) $x ^ { 2 } + 6 x + 36$
B) $\frac { 1 } { x - 6 }$
C) $x ^ { 2 } - 36$
D) $\frac { x ^ { 3 } - 216 } { x - 6 }$

Question 23

Multiply and simplify.
$$\frac { x ^ { 2 } - 5 x - 24 } { 3 x ^ { 2 } - 192 } \cdot \frac { x ^ { 2 } + 16 x + 64 } { 3 x ^ { 2 } + 33 x + 72 }$$

A) $\frac { 1 } { 9 ( x + 8 ) }$
B) 9
C) 0
D) $\frac { 1 } { 9 }$

Question 24

Multiply and simplify.
$$\frac { 10 x - 15 x ^ { 2 } } { 9 x ^ { 2 } + 30 x + 25 } \cdot \frac { 9 x ^ { 2 } + 21 x + 10 } { 9 x ^ { 2 } - 4 }$$

A) $- \frac { 5 x } { 3 x + 5 }$
B) $\frac { 5 x ( 2 - 3 x ) } { 3 x + 5 ) ( 3 x - 2 ) }$
C) $\frac { 5 x } { 3 x + 5 }$
D) 1

Question 25

Multiply and simplify.
$$\frac { x ^ { 3 } + x ^ { 2 } y + 2 x + 2 y } { 3 x ^ { 4 } + 6 x ^ { 2 } } \cdot \frac { 8 x ^ { 3 } } { 4 x ^ { 2 } - 4 y ^ { 2 } }$$

A) $\frac { 2 x } { 3 ( x + y ) }$
B) $\frac { 2 x } { ( x - y ) }$
C) $\frac { 2 x } { 3 ( x - y ) }$
D) $\frac { 2 x ( x + y ) } { 3 ( x - y ) }$

Question 26

Multiply and simplify.
$\frac { 12 x + 12 } { 9 x - 3 } \cdot \frac { 27 x - 9 } { 4 x ^ { 2 } - 4 }$

A) $\frac { 9 } { x - 1 }$
B) $\frac { 3 } { 4 ( x - 1 ) }$
C) $\frac { 3 } { x - 1 }$
D) $\frac { 9 } { ( x - 1 ) ^ { 2 } }$

Question 27

Simplify the rational expression.
$$\frac { 2 x + 2 } { 10 x ^ { 2 } + 16 x + 6 }$$

A) $\frac { 1 } { 5 x + 3 }$
B) $\frac { 2 x + 2 } { 10 x ^ { 2 } + 16 x + 6 }$
C) $\frac { 2 x } { 5 x + 3 }$
D) $\frac { 2 x + 5 } { 5 x + 16 }$

Question 28

Simplify the rational expression.
$$\frac { x ^ { 2 } - 9 } { 3 + x }$$

A) $x + 3$
B) 0
C) $3 - x$
D) $x - 3$

Question 29

Simplify the rational expression.
$\frac { 6 x + 6 } { 2 x ^ { 3 } + 2 }$

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

Question 30

Simplify the rational expression.
$$\frac { x ^ { 2 } - 12 x + 36 } { 6 - x }$$

A) -6
B) $x ^ { 2 } - 6 x$
C) $x - 6$
D) $6 - x$

Question 31

Multiply and simplify.
$$\frac { 3 x - 3 } { 10 } \cdot \frac { 6 } { 1 - x }$$

A) $- \frac { 9 } { 5 }$
B) $- \frac { 18 } { 5 }$
C) $\frac { 9 } { 5 }$
D) $\frac { 9 ( x - 1 ) } { 5 ( 1 - x ) }$

Question 32

Simplify the rational expression.
$$\frac { y ^ { 2 } + 7 y - 18 } { y ^ { 2 } + 4 y - 45 }$$

A) $\frac { 7 y - 18 } { 4 y - 45 }$
B) $\frac { 7 y - 2 } { 4 y - 5 }$
C) $- \frac { y ^ { 2 } + 7 y - 18 } { y ^ { 2 } + 4 y - 45 }$
D) $\frac { y - 2 } { y - 5 }$

Question 33

Multiply and simplify.
$$\frac { x ^ { 2 } - 25 } { 2 } \cdot \frac { x ^ { 2 } + 2 x - 35 } { x ^ { 2 } - 10 x + 25 }$$

A) $\frac { ( x + 5 ) } { 2 }$
B) $( x + 5 ) ( x + 2 )$
C) $\frac { 5 x ^ { 2 } } { 4 }$
D) $\frac { ( x + 5 ) ( x + 7 ) } { 2 }$

Question 34

Simplify the rational expression.
$$\frac { 12 x ^ { 2 } + 36 x ^ { 3 } } { 5 x + 15 x ^ { 2 } }$$

A) $\frac { 12 + 36 x ^ { 3 } } { 5 x + 15 }$
B) $\frac { 12 } { 5 }$
C) $\frac { 12 x ^ { 2 } + 36 x ^ { 3 } } { 5 x + 15 x ^ { 2 } }$
D) $\frac { 12 x } { 5 }$

Question 35

Simplify the rational expression.
$$\frac { x ^ { 3 } - 27 } { 3 - x }$$

A) $\frac { x ^ { 3 } - 27 } { 3 - x }$
B) $x ^ { 2 } - 9$
C) $\frac { 1 } { 3 - x }$
D) $- x ^ { 2 } - 3 x - 9$

Question 36

Simplify the rational expression.
$$\frac { x ^ { 2 } + 4 x + 4 } { x ^ { 2 } + 5 x + 6 }$$

A) $\frac { 4 x + 2 } { 5 x + 3 }$
B) $- \frac { x ^ { 2 } + 4 x + 4 } { x ^ { 2 } + 5 x + 6 }$
C) $\frac { x + 2 } { x + 3 }$
D) $\frac { 4 x + 4 } { 5 x + 6 }$

Question 37

Simplify the rational expression.
$$\frac { x ^ { 2 } - x y + 4 x - 4 y } { x + 4 }$$

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

Question 38

Simplify the rational expression.
$$\frac { 3 x ^ { 2 } - 5 x - 12 } { 9 x ^ { 3 } + 12 x ^ { 2 } + 3 x + 4 }$$

A) 0
B) $\frac { x - 3 } { 3 x ^ { 2 } + 1 }$
C) $\frac { x + 3 } { 3 x ^ { 2 } - 1 }$
D) $- \frac { 6 } { 3 x ^ { 3 } + 6 }$

Question 39

Simplify the rational expression.
$$\frac { 9 x ^ { 2 } - 17 x + 8 } { x - 1 }$$

A) $\frac { 9 x ^ { 2 } - 17 x + 8 } { x - 1 }$
B) $9 x ^ { 2 } - 25$
C) $\frac { 1 } { x - 1 }$
D) $9 x - 8$

Question 40

Multiply and simplify.
$$\frac { 3 x ^ { 3 } - 3 } { 2 x ^ { 2 } + 8 x - 10 } \cdot \frac { 4 x + 8 } { 6 x ^ { 2 } + 6 x + 6 }$$

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

Question 41

Divide and simplify.
$$\frac { 7 b + 14 } { 5 a + 20 } \div \frac { a b - 4 b + 2 a - 8 } { a ^ { 2 } - 8 a + 16 }$$

A) $\frac { 7 ( a - 4 ) } { 5 ( a + 4 ) }$
B) $\frac { 7 ( a + 4 ) } { 5 ( a - 4 ) }$
$C ) - \frac { 7 ( a - 4 ) } { 5 ( a + 4 ) }$
D) $\frac { 5 ( a + 4 ) } { 7 ( a - 4 ) }$

Question 42

Perform the indicated operations.
$$\frac { 5 x ^ { 2 } + 38 x - 63 } { 2 x - 18 } \cdot \frac { x ^ { 2 } - 9 x } { 25 x ^ { 2 } - 49 } \div \frac { 6 x + 54 } { 7 x ^ { 3 } }$$

A) $\frac { 7 x ^ { 4 } } { 12 ( 5 x + 7 ) }$
B) $\frac { 6 ( x + 9 ) ^ { 2 } } { 14 x ^ { 2 } ( 5 x + 7 ) }$
C) $\frac { 7 x ^ { 3 } } { 12 ( 5 x + 7 ) }$
D) $\frac { 12 } { 7 x ^ { 4 } ( 5 x + 7 ) }$

Question 43

Multiply and simplify.
$$\frac { x ^ { 2 } + 10 x + 25 } { x ^ { 2 } + 12 x + 35 } \cdot \frac { x ^ { 2 } + 7 x } { x ^ { 2 } + 7 x + 10 }$$

A) $\frac { x } { x ^ { 2 } + 12 x + 35 }$
B) $\frac { x } { x + 2 }$
C) $\frac { 1 } { x + 2 }$
D) $\frac { x ^ { 2 } + 7 x } { x + 2 }$

Question 44

Perform the indicated operations.
$$\frac { 3 x ^ { 2 } + 8 x + 4 } { y ^ { 2 } + 2 y - 3 } \cdot \frac { y ^ { 2 } + 3 y - 4 } { 12 x ^ { 2 } + 5 x - 2 } \div \frac { 3 x ^ { 2 } + 5 x - 2 } { 8 x ^ { 2 } + 10 x - 3 }$$

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

Question 45

Multiply and simplify.
$$\frac { x ^ { 2 } + 11 x + 30 } { x ^ { 2 } + 14 x + 45 } \cdot \frac { x ^ { 2 } + 9 x } { x ^ { 2 } - 3 x - 54 }$$

A) $\frac { x ^ { 2 } + 9 x } { x - 9 }$
B) $\frac { 1 } { x - 9 }$
C) $\frac { x } { x - 9 }$
D) $\frac { x } { x ^ { 2 } + 14 x + 45 }$

Question 46

Divide and simplify.
$$\frac { 12 x - 3 x ^ { 2 } } { x ^ { 3 } - 64 } \div \frac { x } { x ^ { 2 } + 4 x + 16 }$$

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

Question 47

Find each function value.
If $g ( x ) = \frac { x ^ { 3 } + 2 } { x ^ { 2 } + 2 }$ , find $f ( - 5 ) , f ( 5 )$ , and $f ( 7 )$

A) $- \frac { 41 } { 9 } , \frac { 127 } { 27 } , \frac { 115 } { 17 }$
B) $\frac { 41 } { 9 } , \frac { 127 } { 27 } , - \frac { 115 } { 17 }$
C) $\frac { 41 } { 9 } , \frac { 127 } { 27 } , - \frac { 341 } { 51 }$
D) $- \frac { 127 } { 27 } , \frac { 127 } { 23 } , \frac { 115 } { 17 }$

Question 48

Multiply and simplify.
$$\frac { x ^ { 2 } + 14 x + 48 } { x ^ { 2 } + 8 x + 12 } \cdot \frac { x ^ { 2 } + 11 x + 18 } { x ^ { 2 } + 17 x + 72 }$$

A) 1
B) $\frac { 1 } { x + 9 }$
C) $\frac { x + 8 } { x + 2 }$
D) $\frac { x + 2 } { x + 9 }$

Question 49

Solve the problem.
A publishing company estimates the revenue from the sale of a popular book by the rational function $R ( x ) = \frac { 1040 x ^ { 2 } } { x ^ { 2 } + 4 }$ where $x$ is the number of years since publication and $R ( x )$ is the total revenue in millions of dollars. Find the total revenue at the end of the second year.

A) $\$ 560$ million
B) $\$ 520$ million
C) $\$ 550$ million
D) $\$ 590$ million

Question 50

Divide and simplify.
$$\frac { x ^ { 2 } - 9 } { 4 x + 12 } \div \frac { 4 x ^ { 2 } - 24 x + 36 } { x ^ { 2 } + 6 x + 9 }$$

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

Question 51

Find each function value.
If $f ( x ) = \frac { x + 7 } { 3 x - 2 }$ , find $f ( 0 ) , f ( 2 )$ , and $f ( 5 )$

A) $- \frac { 7 } { 2 } , \frac { 9 } { 8 } , - \frac { 12 } { 13 }$
B) $- \frac { 7 } { 2 } , - \frac { 2 } { 3 } , - \frac { 12 } { 13 }$
C) $\frac { 7 } { 2 } , - \frac { 9 } { 8 } , \frac { 12 } { 13 }$
D) $- \frac { 7 } { 2 } , - \frac { 9 } { 8 } , \frac { 12 } { 13 }$

Question 52

Divide and simplify.
$$\frac { x ^ { 2 } + 6 x + 9 } { x ^ { 2 } + 10 x + 21 } \div \frac { x ^ { 2 } + 3 x } { x ^ { 2 } + 12 x + 35 }$$

A) $\frac { x + 5 } { x }$
B) $x + 5$
C) $\frac { x } { x ^ { 2 } + 10 x + 21 }$
D) $\frac { x + 5 } { x ^ { 2 } + 7 x }$

Question 53

Divide and simplify.
$$\frac { x ^ { 2 } - 2 x - 15 } { 2 x ^ { 2 } - 50 } \div \frac { x ^ { 2 } + 8 x + 15 } { x ^ { 2 } + 10 x + 25 }$$

A) $\frac { 1 } { 2 }$
B) $- 2$
C) 2
D) $\frac { 2 ( x + 5 ) ^ { 2 } } { ( x + 3 ) ^ { 2 } }$

Question 54

Divide and simplify.
$$\frac { x - y } { x y } \div \frac { x ^ { 2 } - y ^ { 2 } } { 4 x ^ { 2 } y }$$

A) $\frac { 4 } { y }$
B) $- \frac { 4 } { y }$
C) $\frac { 4 x } { x - y }$
D) $\frac { 4 x } { x + y }$

Question 55

Multiply and simplify.
$$\frac { x ^ { 2 } - 5 x + 6 } { x ^ { 2 } - 17 x + 60 } \cdot \frac { x ^ { 2 } - 13 x + 40 } { x ^ { 2 } - 9 x + 14 }$$

A) $\frac { ( x - 3 ) ( x - 8 ) } { ( x - 12 ) ( x - 7 ) }$
B) $\frac { \left( x ^ { 2 } - 5 x + 6 \right) \left( x ^ { 2 } - 13 x + 40 \right) } { \left( x ^ { 2 } - 17 x + 60 \right) \left( x ^ { 2 } - 9 x + 14 \right) }$
C) $\frac { ( x + 3 ) ( x + 8 ) } { ( x + 12 ) ( x + 7 ) }$
D) $\frac { ( x - 3 ) } { ( x - 7 ) }$

Question 56

Divide and simplify.
$$\frac { 2 x ^ { 2 } - 4 x - 16 } { 5 x ^ { 2 } - 30 x + 40 } \div \frac { x ^ { 2 } - 6 x + 8 } { x ^ { 2 } - 4 x + 4 }$$

A) $\frac { 5 ( x - 4 ) } { 2 ( x + 2 ) }$
B) $\frac { 5 ( x - 2 ) } { 2 ( x + 4 ) }$
C) $\frac { 2 ( x - 2 ) } { 5 ( x + 4 ) }$
D) $\frac { 2 ( x + 2 ) } { 5 ( x - 4 ) }$

Question 57

Multiply and simplify.
$\frac { 90 x y ^ { 2 } } { x ^ { 2 n } - 36 } \cdot \frac { 10 x ^ { n } - 60 } { 9 x ^ { 2 } y ^ { 2 } }$

A) $\frac { 54 } { x \left( x ^ { n } + 6 \right) }$
B) $\frac { 10 } { x \left( x ^ { n } + 6 \right) }$
C) $\frac { 100 } { x \left( x ^ { n } + 6 \right) }$
D) $\frac { 100 x } { x ^ { n } + 6 }$

Question 58

Divide and simplify.
$$\frac { 4 } { 3 x } \div \frac { 16 - 4 x } { 48 - 12 x }$$

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

Question 59

Multiply and simplify.
$$\frac { x ^ { 3 } + 1 } { x ^ { 3 } - x ^ { 2 } + x } \cdot \frac { 4 x } { - 28 x - 28 }$$

A) $- \frac { x ^ { 2 } + 1 } { 7 }$
B) $- \frac { 1 } { 7 }$
C) $\frac { x + 1 } { 7 ( - x - 1 ) }$
D) $- \frac { x ^ { 3 } + 1 } { 7 ( x + 1 ) }$

Question 60

Divide and simplify.
$$\frac { x ^ { 2 } - 10 x + 25 } { 12 x - 60 } \div \frac { 9 x - 45 } { 108 }$$

A) 108
B) 1
C) $\frac { ( x - 5 ) ^ { 2 } } { 144 }$
D) $\frac { x ^ { 2 } - 10 x + 25 } { ( x - 5 ) ^ { 2 } }$

Question 61

Perform the indicated operation. Simplify if possible.
$$\frac { x - 2 } { 2 x } - \frac { x + 2 } { 2 x }$$

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

Question 62

Perform the indicated operation. Simplify if possible.
$$\frac { x ^ { 2 } - 8 x } { x - 2 } + \frac { 12 } { x - 2 }$$

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

Question 63

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 $\mathrm { C } ( \mathrm { t } ) = \frac { 4 \mathrm { t } } { 2 \mathrm { t } ^ { 2 } + 2 }$ . Estimate the drug's concentration after 2 hours. (Round to the nearest hundredth.)

A) $0.69$ milligrams
B) $1.33$ milligrams
C) $0.80$ milligrams
D) $1.22$ milligrams

Question 64

Perform the indicated operation. Simplify if possible.
$$\frac { 3 } { 9 x ^ { 2 } } + \frac { 3 } { 9 x ^ { 2 } }$$

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

Question 65

Perform the indicated operation. Simplify if possible.
$$\frac { 14 } { 9 x ^ { 2 } } - \frac { 8 } { 9 x ^ { 2 } }$$

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

Question 66

Perform the indicated operation. Simplify if possible.
$$\frac { 13 } { 9 x ^ { 2 } y } - \frac { 6 } { 9 x ^ { 2 } y }$$

A) 7
B) $\frac { 7 } { 18 x ^ { 2 } y }$
C) $\frac { 9 x ^ { 2 } y } { 7 }$
D) $\frac { 7 } { 9 x ^ { 2 } y }$

Question 67

Solve the problem.
To calculate the drug dosage for a child, a pharmacist may use the formula $\mathrm { d } ( \mathrm { x } ) = \frac { \mathrm { Dx } } { \mathrm { x } + 9 } , 0 \leq \mathrm { x } \leq 12$ . 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 $10 \mathrm { mg }$ ? (Round to the nearest tenth.)

A) $0.8 \mathrm { mg }$
B) $3.3 \mathrm { mg }$
C) $2.5 \mathrm { mg }$
D) $10.0 \mathrm { mg }$

Question 68

Find the LCD of the rational expressions in the list.
$$\frac { 1 } { 63 x } , \frac { 1 } { 9 x ^ { 2 } } , \frac { 1 } { 7 x ^ { 3 } }$$

A) $63 x ^ { 2 }$
B) $63 x ^ { 3 }$
C) $7 x ^ { 3 }$
D) $63 x ^ { 5 }$

Question 69

Perform the indicated operation. Simplify if possible.
$$\frac { x ^ { 2 } } { x - 1 } - \frac { 1 } { x - 1 }$$

A) 0
B) $x ^ { 2 } - 1$
C) $x - 1$
D) $x + 1$

Question 70

Perform the indicated operation. Simplify if possible.
$$\frac { x } { x + 7 } + \frac { - 7 } { x + 7 }$$

A) $\frac { - 7 x } { x ^ { 2 } + 14 x + 49 }$
B) $- 1$
C) $\frac { - 6 x } { x + 7 }$
D) $\frac { x - 7 } { x + 7 }$

Question 71

Solve the problem.
Find the area of the triangle.

A) $\frac { 1 } { 12 y }$ sq. in.
B) $\frac { 1 } { 3 y } s q$ . in.
C) $\frac { 1 } { 6 y } s q .$ in.
D) $\frac { 5 } { 3 y } s q$ . in.

Question 72

Solve the problem.
A company that produces radios has costs given by the function $\mathrm { C } ( \mathrm { x } ) = 30 \mathrm { x } + 15,000$ , 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 $\bar { C } ( \mathrm { x } ) = \frac { 30 \mathrm { x } + 15,000 } { \mathrm { x } }$ . Find $\bar { C } ( 250 )$ . (Round to the nearest dollar, if necessary.)

A) $\$ 33$
B) $\$ 120$
C) $\$ 90$
D) $\$ 36$

Question 73

Perform the indicated operation. Simplify if possible.
$$\frac { 4 } { 10 x z ^ { 2 } } + \frac { 3 } { 10 x z ^ { 2 } }$$

A) $\frac { 10 \mathrm { xz } ^ { 2 } } { 7 }$
B) 1
C) $\frac { 7 } { 20 \mathrm { xz } ^ { 2 } }$
D) $\frac { 7 } { 10 \mathrm { xz } ^ { 2 } }$

Question 74

Find the LCD of the rational expressions in the list.
$$\frac { 1 } { 10 x y } , \frac { y ^ { 3 } } { 25 x ^ { 2 } }$$

A) $50 x y ^ { 2 }$
B) $50 x ^ { 2 } y$
C) $50 x y ^ { 3 }$
D) $50 x ^ { 3 } y$

Question 75

Perform the indicated operation. Simplify if possible.
$\frac { 5 } { 8 x ^ { 2 } } + \frac { 5 } { 8 x ^ { 2 } }$

A) $\frac { 4 } { 5 x ^ { 2 } }$
B) $\frac { 5 } { 2 x ^ { 4 } }$
C) 5
D) $\frac { 5 } { 4 x ^ { 2 } }$

Question 76

Solve the problem.
Find the area of the rectangle.

A) $\frac { 5 } { 2 x ^ { 2 } ( x - 9 ) } s q . f t$
B) $\frac { 5 } { 2 x ^ { 2 } ( x + 9 ) } s q . f t$
C) $\frac { 5 x ^ { 2 } } { 2 ( x - 9 ) } s q . f t$
D) $\frac { 9 } { 2 x ^ { 3 } - 18 }$ sq. $\mathrm { ft }$

Question 77

Perform the indicated operation. Simplify if possible.
$$\frac { 4 x ^ { 2 } } { x - 1 } + \frac { - 4 x } { x - 1 }$$

A) $\frac { 4 x } { x - 1 }$
B) $4 \mathrm { x }$
C) 0
D) $\frac { 4 x ( x + 1 ) } { x - 1 }$

Question 78

Solve the problem.
A parallelogram has area $\frac { x ^ { 2 } - 12 x + 35 } { x }$ square feet and height $\frac { x ^ { 6 } } { x - 7 }$ feet. Express the length of its base as a ration expression in $x$ .

A) $\frac { x - 5 } { x ^ { 5 } } \mathrm { ft }$
B) $\frac { ( x - 5 ) ( x - 7 ) ^ { 2 } } { x ^ { 7 } } \mathrm { ft }$
C) $\frac { ( x - 5 ) ( x - 7 ) ^ { 2 } } { x ^ { 5 } } \mathrm { ft }$
D) $x ^ { 5 } ( x - 5 ) f t$

Question 79

Perform the indicated operation. Simplify if possible.
$$\frac { 10 x - 4 } { x ^ { 2 } + 10 x + 21 } + \frac { 7 - 9 x } { x ^ { 2 } + 10 x + 21 }$$

A) $\frac { 1 } { x + 7 }$
B) $\frac { 1 } { x + 3 }$
C) $\frac { 1 } { x ^ { 2 } + 10 x + 21 }$
D) $\frac { x - 3 } { x ^ { 2 } + 10 x + 21 }$

Question 80

Find the LCD of the rational expressions in the list.
$\frac { 5 } { t } , \frac { - 9 } { t - 5 }$

A) 5
B) $t ( t - 5 )$
C) $t - 5$
D) $- 5$