Question 1

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

Accepted Answer

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)  $\frac { ( x - 7 ) ^ { 2 } } { 90 ( x - y ) ^ { 2 } }$ 
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 2

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

Accepted Answer

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

Question 3

Perform the indicated operations. Simplify the result, if possible.
-$$\frac { x } { x ^ { 2 } - 25 } + \frac { 5 } { x + 5 } - \frac { 6 } { x }$$

Accepted Answer

A)  $\frac { 6 x ^ { 2 } - 25 x + 150 } { x ( x + 5 ) ( x - 5 ) }$ 
B)  $\frac { 25 ( x + 6 ) } { x ( x + 5 ) ( x - 5 ) }$ 
C)  $\frac { 25 ( x - 6 ) } { ( x + 5 ) ( x - 5 ) }$ 
D)  $\frac { - 25 ( x - 6 ) } { x ( x + 5 ) ( x - 5 ) }$ 
A)  $\frac { 6 x ^ { 2 } - 25 x + 150 } { x ( x + 5 ) ( x - 5 ) }$ 
B)  $\frac { 25 ( x + 6 ) } { x ( x + 5 ) ( x - 5 ) }$ 
C)  $\frac { 25 ( x - 6 ) } { ( x + 5 ) ( x - 5 ) }$ 
D)  $\frac { - 25 ( x - 6 ) } { x ( x + 5 ) ( x - 5 ) }$

Question 4

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

Accepted Answer

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

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?

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 6

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.

Accepted Answer

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

Question 7

Simplify the complex fraction.
-$$\frac { \frac { 7 } { 3 r - 1 } - 7 } { \frac { 7 } { 3 r - 1 } + 7 }$$

Accepted Answer

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

Question 8

Simplify the complex fraction.
-$$\frac { \frac { 1 } { k + 6 } } { \frac { 5 } { k ^ { 2 } - 36 } }$$

Accepted Answer

A)  $\frac { k - 6 } { 5 }$ 
B)  $k - 6$ 
C)  $\frac { 5 } { k - 6 }$ 
D)  $\frac { \mathrm { k } + 6 } { 5 }$ 
A)  $\frac { k - 6 } { 5 }$ 
B)  $k - 6$ 
C)  $\frac { 5 } { k - 6 }$ 
D)  $\frac { \mathrm { k } + 6 } { 5 }$

Question 9

Perform the indicated operations. Simplify the result, if possible.
-$$\frac { 10 } { x ^ { 2 } + 5 x } + \frac { 6 } { x } + \frac { 2 } { x + 5 }$$

Accepted Answer

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

Question 10

Simplify the complex fraction.
-$$\frac { \frac { 16 p ^ { 2 } - 49 q ^ { 2 } } { p q } } { \frac { 4 } { q } - \frac { 7 } { p } }$$

Accepted Answer

A)  $\frac { 7 p + 4 q } { p q }$ 
B)  $\frac { p q } { 4 p + 7 q }$ 
C)  $4 p + 7 q$ 
D)  $7 p + 4 q$ 
A)  $\frac { 7 p + 4 q } { p q }$ 
B)  $\frac { p q } { 4 p + 7 q }$ 
C)  $4 p + 7 q$ 
D)  $7 p + 4 q$

Question 11

Simplify the complex fraction.
-$\frac { \frac { x } { 2 } } { \frac { 3 } { x + 4 } }$

Accepted Answer

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

Question 12

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

Accepted Answer

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 $f : ( \infty , 2 ) \cup ( 2 , \infty )$ 
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 $f : ( \infty , 2 ) \cup ( 2 , \infty )$

Question 13

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

Accepted Answer

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)  $\frac { x ^ { 2 } + 3 x + 16 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
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 14

Simplify the complex fraction.
-$\frac { \frac { 3 } { y } } { \frac { 2 } { y + 8 } }$

Accepted Answer

A)  $6 y ( y + 8 )$ 
B)  $\frac { 3 ( y + 8 ) } { 2 y }$ 
C)  $\frac { y + 8 } { 6 y }$ 
D)  $\frac { 2 y } { 3 ( y + 8 ) }$ 
A)  $6 y ( y + 8 )$ 
B)  $\frac { 3 ( y + 8 ) } { 2 y }$ 
C)  $\frac { y + 8 } { 6 y }$ 
D)  $\frac { 2 y } { 3 ( y + 8 ) }$

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

Solve the problem.
-Write the average of $\frac { 1 } { \mathrm { x } ^ { 3 } }$ and $\frac { 10 } { \mathrm { x } }$ as a simplified rational expression. (Hint: To find the average of two numbers, find their sum and divide by 2 .)

Accepted Answer

A)  $\frac { 2 + 20 x ^ { 2 } } { x ^ { 3 } }$ 
B)  $\frac { 1 + 10 x ^ { 3 } } { 2 x ^ { 4 } }$ 
C)  $\frac { 1 + 10 x ^ { 2 } } { 2 x ^ { 3 } }$ 
D)  $\frac { 11 } { 2 x ^ { 3 } + 2 x }$ 
A)  $\frac { 2 + 20 x ^ { 2 } } { x ^ { 3 } }$ 
B)  $\frac { 1 + 10 x ^ { 3 } } { 2 x ^ { 4 } }$ 
C)  $\frac { 1 + 10 x ^ { 2 } } { 2 x ^ { 3 } }$ 
D)  $\frac { 11 } { 2 x ^ { 3 } + 2 x }$

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

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)  $\frac { 8 x - 12 } { x ( x - 2 ) }$ 
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 20

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

Accepted Answer

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

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

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

Perform the indicated operations. Simplify the result, if possible. - $\frac { x } { x ^ { 2 } - 25 } + \frac { 5 } { x + 5 } - \frac { 6 } { x }$

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

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?

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.

Simplify the complex fraction. - $\frac { \frac { 7 } { 3 r - 1 } - 7 } { \frac { 7 } { 3 r - 1 } + 7 }$

Simplify the complex fraction. - $\frac { \frac { 1 } { k + 6 } } { \frac { 5 } { k ^ { 2 } - 36 } }$

Perform the indicated operations. Simplify the result, if possible. - $\frac { 10 } { x ^ { 2 } + 5 x } + \frac { 6 } { x } + \frac { 2 } { x + 5 }$

Simplify the complex fraction. - $\frac { \frac { 16 p ^ { 2 } - 49 q ^ { 2 } } { p q } } { \frac { 4 } { q } - \frac { 7 } { p } }$

Simplify the complex fraction. - $\frac { \frac { x } { 2 } } { \frac { 3 } { x + 4 } }$

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

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

Simplify the complex fraction. - $\frac { \frac { 3 } { y } } { \frac { 2 } { y + 8 } }$

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

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

Solve the problem. -Write the average of $\frac { 1 } { \mathrm { x } ^ { 3 } }$ and $\frac { 10 } { \mathrm { x } }$ as a simplified rational expression. (Hint: To find the average of two numbers, find their sum and divide by 2 .)

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

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

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

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Exam 6: Rational Expressions, Functions, and Equations

Divide as indicated. - x2−7x+xy−7y9x2−9y2÷x−710x−10y\frac { x ^ { 2 } - 7 x + x y - 7 y } { 9 x ^ { 2 } - 9 y ^ { 2 } } \div \frac { x - 7 } { 10 x - 10 y }9x2−9y2x2−7x+xy−7y​÷10x−10yx−7​

Divide as indicated. - (y−5)23÷3y−159\frac { ( y - 5 ) ^ { 2 } } { 3 } \div \frac { 3 y - 15 } { 9 }3(y−5)2​÷93y−15​

Perform the indicated operations. Simplify the result, if possible. - xx2−25+5x+5−6x\frac { x } { x ^ { 2 } - 25 } + \frac { 5 } { x + 5 } - \frac { 6 } { x }x2−25x​+x+55​−x6​

Simplify the rational expression. If the rational expression cannot be simplified, so state. - 3x+212x2+23x+10\frac { 3 x + 2 } { 12 x ^ { 2 } + 23 x + 10 }12x2+23x+103x+2​

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?

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.

Simplify the complex fraction. - 73r−1−773r−1+7\frac { \frac { 7 } { 3 r - 1 } - 7 } { \frac { 7 } { 3 r - 1 } + 7 }3r−17​+73r−17​−7​

Simplify the complex fraction. - 1k+65k2−36\frac { \frac { 1 } { k + 6 } } { \frac { 5 } { k ^ { 2 } - 36 } }k2−365​k+61​​

Perform the indicated operations. Simplify the result, if possible. - 10x2+5x+6x+2x+5\frac { 10 } { x ^ { 2 } + 5 x } + \frac { 6 } { x } + \frac { 2 } { x + 5 }x2+5x10​+x6​+x+52​

Simplify the complex fraction. - 16p2−49q2pq4q−7p\frac { \frac { 16 p ^ { 2 } - 49 q ^ { 2 } } { p q } } { \frac { 4 } { q } - \frac { 7 } { p } }q4​−p7​pq16p2−49q2​​

Simplify the complex fraction. - x23x+4\frac { \frac { x } { 2 } } { \frac { 3 } { x + 4 } }x+43​2x​​

Find the domain of the rational function. - f(x)=2x−2f ( x ) = \frac { 2 } { x - 2 }f(x)=x−22​

Perform the indicated operations. Simplify the result, if possible. - xx2−16−4x2+5x+4\frac { x } { x ^ { 2 } - 16 } - \frac { 4 } { x ^ { 2 } + 5 x + 4 }x2−16x​−x2+5x+44​

Simplify the complex fraction. - 3y2y+8\frac { \frac { 3 } { y } } { \frac { 2 } { y + 8 } }y+82​y3​​

Divide as indicated. - x2−20x+1006x−60÷3x−3018\frac { x ^ { 2 } - 20 x + 100 } { 6 x - 60 } \div \frac { 3 x - 30 } { 18 }6x−60x2−20x+100​÷183x−30​

Find the function value. - f(x)=x2+4x3−7x;f(−5)f ( x ) = \frac { x ^ { 2 } + 4 } { x ^ { 3 } - 7 x } ; f ( - 5 )f(x)=x3−7xx2+4​;f(−5)

Solve the problem. -Write the average of 1x3\frac { 1 } { \mathrm { x } ^ { 3 } }x31​ and 10x\frac { 10 } { \mathrm { x } }x10​ as a simplified rational expression. (Hint: To find the average of two numbers, find their sum and divide by 2 .)

Divide as indicated. - 4x25÷x330\frac { 4 x ^ { 2 } } { 5 } \div \frac { x ^ { 3 } } { 30 }54x2​÷30x3​

Perform the indicated operations. Simplify the result, if possible. - 4x+8x−2\frac { 4 } { x } + \frac { 8 } { x - 2 }x4​+x−28​

Multiply as indicated. - 2x33⋅9x2\frac { 2 x ^ { 3 } } { 3 } \cdot \frac { 9 } { x ^ { 2 } }32x3​⋅x29​