Question 1

Determine the equations of any horizontal and vertical asymptotes of
$$f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 16 } .$$

Accepted Answer

A)  horizontal: $y = 4$; vertical: $x = - 4$ 
B)  horizontal: $x = 1$; vertical: none 
C)  horizontal: $y = - 4$; vertical: $x = 1$ 
D)  horizontal: $y = 1$; vertical: none 
E)  horizontal: none; vertical: none 
A)  horizontal: $y = 4$; vertical: $x = - 4$ 
B)  horizontal: $x = 1$; vertical: none 
C)  horizontal: $y = - 4$; vertical: $x = 1$ 
D)  horizontal: $y = 1$; vertical: none 
E)  horizontal: none; vertical: none

Question 2

Find the zeros of the function below algebraically, if any exist.
$$f ( x ) = x ^ { 6 } - 9 x ^ { 3 } + 8$$

Accepted Answer

A)  1 and 2 
B)  $- 4$ and 1 
C)  $- 4$ and 2 
D)  $- 4 , - 1,1$, and 4 
E)  $- 4 , - 2,2$, and 4 
A)  1 and 2 
B)  $- 4$ and 1 
C)  $- 4$ and 2 
D)  $- 4 , - 1,1$, and 4 
E)  $- 4 , - 2,2$, and 4

Question 3

Write the complex conjugate of the complex number $3 - \sqrt { 2 } i$.

Accepted Answer

A)  $- 3 - \sqrt { 2 } i$ 
B)  $3 - \sqrt { - 2 } i$ 
C)  $- 3 - \sqrt { - 2 } i$ 
D)  $3 + \sqrt { 2 } i$ 
E)  $- 3 + \sqrt { 2 } i$ 
A)  $- 3 - \sqrt { 2 } i$ 
B)  $3 - \sqrt { - 2 } i$ 
C)  $- 3 - \sqrt { - 2 } i$ 
D)  $3 + \sqrt { 2 } i$ 
E)  $- 3 + \sqrt { 2 } i$

Question 4

Use long division to divide.
$$\left( x ^ { 4 } - x ^ { 2 } - 5 \right) \div \left( x ^ { 2 } + 4 x - 1 \right)$$

Accepted Answer

A)  $x ^ { 2 } - 4 x + 4$ 
B)  $x ^ { 2 } + 4 x - 4$ 
C)  $x ^ { 2 } - 4 x + 16 + \frac { - 68 x + 11 } { x ^ { 2 } + 4 x - 1 }$ 
D)  $x ^ { 2 } + 4 x - 4 + \frac { - 5 x - 1 } { x ^ { 2 } + 4 x - 1 }$ 
E)  $x ^ { 2 } - 4 x + 4 - \frac { 4 } { x ^ { 2 } - 4 x + 4 }$ 
A)  $x ^ { 2 } - 4 x + 4$ 
B)  $x ^ { 2 } + 4 x - 4$ 
C)  $x ^ { 2 } - 4 x + 16 + \frac { - 68 x + 11 } { x ^ { 2 } + 4 x - 1 }$ 
D)  $x ^ { 2 } + 4 x - 4 + \frac { - 5 x - 1 } { x ^ { 2 } + 4 x - 1 }$ 
E)  $x ^ { 2 } - 4 x + 4 - \frac { 4 } { x ^ { 2 } - 4 x + 4 }$

Question 5

Given $f ( x ) = \frac { 5 x + 4 } { 5 x ^ { 2 } + 4 x }$. Determine the domain of $f ( x )$ and find any vertical asymptotes.

Accepted Answer

A)  domain: all real numbers except $x = - \frac { 4 } { 5 }$ vertical asymptote: $x = 0$ 
B)  domain: all real numbers except $x = 0$ and $x = - \frac { 4 } { 5 }$
vertical asymptote: $x = 0$ 
C)  domain: all real numbers
vertical asymptotes: $x = 0$ and $x = - \frac { 4 } { 5 }$ 
D)  domain: all real numbers except $x = 0$ and $x = \frac { 4 } { 5 }$ vertical asymptote: $x = 0$ 
E)  domain: all real numbers except $x = \frac { 4 } { 5 }$ vertical asymptotes: $x = 0$ and $x = - \frac { 4 } { 5 }$ 
A)  domain: all real numbers except $x = - \frac { 4 } { 5 }$ vertical asymptote: $x = 0$ 
B)  domain: all real numbers except $x = 0$ and $x = - \frac { 4 } { 5 }$
vertical asymptote: $x = 0$ 
C)  domain: all real numbers
vertical asymptotes: $x = 0$ and $x = - \frac { 4 } { 5 }$ 
D)  domain: all real numbers except $x = 0$ and $x = \frac { 4 } { 5 }$ vertical asymptote: $x = 0$ 
E)  domain: all real numbers except $x = \frac { 4 } { 5 }$ vertical asymptotes: $x = 0$ and $x = - \frac { 4 } { 5 }$

Question 6

Simplify $\frac { 1 - 2 i } { 5 i }$ and write the answer in standard form.

Accepted Answer

A)  $\frac { 2 } { 5 } - \frac { i } { 5 }$ 
B)  $- \frac { 2 } { 5 } - \frac { i } { 5 }$ 
C)  $- \frac { 2 } { 5 } + \frac { i } { 5 }$ 
D)  $\frac { 1 } { 5 } - \frac { 2 i } { 5 }$ 
E)  $- \frac { 1 } { 5 } - \frac { 2 i } { 5 }$ 
A)  $\frac { 2 } { 5 } - \frac { i } { 5 }$ 
B)  $- \frac { 2 } { 5 } - \frac { i } { 5 }$ 
C)  $- \frac { 2 } { 5 } + \frac { i } { 5 }$ 
D)  $\frac { 1 } { 5 } - \frac { 2 i } { 5 }$ 
E)  $- \frac { 1 } { 5 } - \frac { 2 i } { 5 }$

Question 7

Simplify $( 3 - 6 i ) ^ { 2 } - ( 3 + 6 i ) ^ { 2 }$ and write the answer in standard form.

Accepted Answer

A)  0 
B)  $- 72 i$ 
C)  $18 - 72 i$ 
D)  $18 + 72 i$ 
E)  $6 - 24 i$ 
A)  0 
B)  $- 72 i$ 
C)  $18 - 72 i$ 
D)  $18 + 72 i$ 
E)  $6 - 24 i$

Question 8

Find two positive real numbers whose product is a maximum and whose sum is 146 .

Accepted Answer

A)  71,75 
B)  73,73 
C)  78,68 
D)  82,64 
E)  61,85 
A)  71,75 
B)  73,73 
C)  78,68 
D)  82,64 
E)  61,85

Question 9

Determine the zeros (if any) of the rational function $g ( x ) = \frac { x ^ { 3 } - 1 } { x ^ { 2 } + 5 }$.

Accepted Answer

A)  $x = - 1 , x = 1$ 
B)  $x = 1$ 
C)  $x = - \sqrt { 5 } , x = \sqrt { 5 } , x = 1$ 
D)  $x = - \sqrt { 5 } , x = \sqrt { 5 } , x = - 1 , x = 1$ 
E)  no zeros 
A)  $x = - 1 , x = 1$ 
B)  $x = 1$ 
C)  $x = - \sqrt { 5 } , x = \sqrt { 5 } , x = 1$ 
D)  $x = - \sqrt { 5 } , x = \sqrt { 5 } , x = - 1 , x = 1$ 
E)  no zeros

Question 10

Describe the right-hand and the left-hand behavior of the graph of $q ( x ) = 7 x ^ { 5 } + x ^ { 3 } + 7$

Accepted Answer

A)  Because the degree is odd and the leading coefficient is positive, the graph falls to the left and falls to the right. 
B)  Because the degree is odd and the leading coefficient is positive, the graph rises to the left and falls to the right. 
C)  Because the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right. 
D)  Because the degree is odd and the leading coefficient is positive, the graph rises to the left and rises to the right. 
E)  Because the degree is even and the leading coefficient is positive, the graph rises to the left and rises to the right. 
A)  Because the degree is odd and the leading coefficient is positive, the graph falls to the left and falls to the right. 
B)  Because the degree is odd and the leading coefficient is positive, the graph rises to the left and falls to the right. 
C)  Because the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right. 
D)  Because the degree is odd and the leading coefficient is positive, the graph rises to the left and rises to the right. 
E)  Because the degree is even and the leading coefficient is positive, the graph rises to the left and rises to the right.

Question 11

Determine the zeros (if any) of the rational function $g ( x ) = 7 + \frac { 2 } { x ^ { 2 } + 7 }$.

Accepted Answer

A)  $x = - \sqrt { 7 } , x = \sqrt { 7 }$ 
B)  $x = - 2$ 
C)  $x = - \frac { 2 } { 7 } , x = \frac { 2 } { 7 }$ 
D)  $x = - 7 , x = 7$ 
E)  no zeros 
A)  $x = - \sqrt { 7 } , x = \sqrt { 7 }$ 
B)  $x = - 2$ 
C)  $x = - \frac { 2 } { 7 } , x = \frac { 2 } { 7 }$ 
D)  $x = - 7 , x = 7$ 
E)  no zeros

Question 12

Determine the value that $f ( x ) = \frac { 4 x - 6 } { x ^ { 2 } - 7 }$ approaches as $x$ increases and decreases in magnitude without bound.

Accepted Answer

A)  8 
B)  6 
C)  4 
D)  2 
E)  0 
A)  8 
B)  6 
C)  4 
D)  2 
E)  0

Question 13

Simplify $\frac { 2 + 5 i } { 3 i }$ and write the answer in standard form.

Accepted Answer

A)  $- \frac { 5 } { 3 } - \frac { 2 i } { 3 }$ 
B)  $\frac { 5 } { 3 } - \frac { 2 i } { 3 }$ 
C)  $\frac { 5 } { 3 } + \frac { 2 i } { 3 }$ 
D)  $\frac { 2 } { 3 } + \frac { 5 i } { 3 }$ 
E)  $- \frac { 2 } { 3 } + \frac { 5 i } { 3 }$ 
A)  $- \frac { 5 } { 3 } - \frac { 2 i } { 3 }$ 
B)  $\frac { 5 } { 3 } - \frac { 2 i } { 3 }$ 
C)  $\frac { 5 } { 3 } + \frac { 2 i } { 3 }$ 
D)  $\frac { 2 } { 3 } + \frac { 5 i } { 3 }$ 
E)  $- \frac { 2 } { 3 } + \frac { 5 i } { 3 }$

Question 14

Find all zeros of the function $f ( x ) = ( x + 6 ) ( x + 3 i ) ( x - 3 i )$.

Accepted Answer

A)  $x = 6 , - 3 i , 3 i$ 
B)  $x = - 6,3 i$ 
C)  $x = - 6 , - 3,3$ 
D)  $x = - 6 , - 3 i , 3 i$ 
E)  $x = - 6$ 
A)  $x = 6 , - 3 i , 3 i$ 
B)  $x = - 6,3 i$ 
C)  $x = - 6 , - 3,3$ 
D)  $x = - 6 , - 3 i , 3 i$ 
E)  $x = - 6$

Question 15

Simplify $\frac { 4 + 3 i } { 5 + 2 i }$ and write the answer in standard form.

Accepted Answer

A)  $- \frac { 26 } { 29 } + \frac { 7 } { 29 } i$ 
B)  $\frac { 26 } { 29 } - \frac { 7 } { 29 } i$ 
C)  $\frac { 26 } { 29 } + \frac { 7 } { 29 } i$ 
D)  $\frac { 7 } { 29 } + \frac { 26 } { 29 } i$ 
E)  $\frac { 7 } { 29 } - \frac { 26 } { 29 } i$ 
A)  $- \frac { 26 } { 29 } + \frac { 7 } { 29 } i$ 
B)  $\frac { 26 } { 29 } - \frac { 7 } { 29 } i$ 
C)  $\frac { 26 } { 29 } + \frac { 7 } { 29 } i$ 
D)  $\frac { 7 } { 29 } + \frac { 26 } { 29 } i$ 
E)  $\frac { 7 } { 29 } - \frac { 26 } { 29 } i$

Question 16

Simplify $( - 4 + i ) ( 4 - 5 i )$ and write the answer in standard form.

Accepted Answer

A)  $16 + 24 i$ 
B)  $- 11 - 24 i$ 
C)  $- 16 + 24 i$ 
D)  $- 16 - 21 i$ 
E)  $- 11 + 24 i$ 
A)  $16 + 24 i$ 
B)  $- 11 - 24 i$ 
C)  $- 16 + 24 i$ 
D)  $- 16 - 21 i$ 
E)  $- 11 + 24 i$

Question 17

Find a fifth degree polynomial function of the lowest degree that has the zeros below and whose leading coefficient is one.
$- 3 , - 1,0,1,3$

Accepted Answer

A)  $f ( x ) = x ^ { 5 } + 7 x ^ { 4 } - 19 x ^ { 3 } - 32 x ^ { 2 } + 48 x$ 
B)  $f ( x ) = x ^ { 5 } + 7 x ^ { 4 } - 19 x ^ { 3 } + 32 x ^ { 2 } + 48 x$ 
C)  $f ( x ) = x ^ { 5 } + 4 x ^ { 4 } - 13 x ^ { 3 } + 3 x ^ { 2 } + 12 x$ 
D)  $f ( x ) = x ^ { 5 } + 5 x ^ { 4 } - 13 x ^ { 3 } + 27 x ^ { 2 } + 36 x$ 
E)  $f ( x ) = x ^ { 5 } - 10 x ^ { 3 } + 9 x$ 
A)  $f ( x ) = x ^ { 5 } + 7 x ^ { 4 } - 19 x ^ { 3 } - 32 x ^ { 2 } + 48 x$ 
B)  $f ( x ) = x ^ { 5 } + 7 x ^ { 4 } - 19 x ^ { 3 } + 32 x ^ { 2 } + 48 x$ 
C)  $f ( x ) = x ^ { 5 } + 4 x ^ { 4 } - 13 x ^ { 3 } + 3 x ^ { 2 } + 12 x$ 
D)  $f ( x ) = x ^ { 5 } + 5 x ^ { 4 } - 13 x ^ { 3 } + 27 x ^ { 2 } + 36 x$ 
E)  $f ( x ) = x ^ { 5 } - 10 x ^ { 3 } + 9 x$

Question 18

Determine the zeros (if any) of the rational function $g ( x ) = 7 + \frac { 3 } { x ^ { 2 } + 7 }$.

Accepted Answer

A)  $\quad x = - \sqrt { 7 } , x = \sqrt { 7 }$ 
B)  $x = - 3$ 
C)  $x = - \frac { 3 } { 7 } , x = \frac { 3 } { 7 }$ 
D)  $x = - 7 , x = 7$ 
E)  no zeros 
A)  $\quad x = - \sqrt { 7 } , x = \sqrt { 7 }$ 
B)  $x = - 3$ 
C)  $x = - \frac { 3 } { 7 } , x = \frac { 3 } { 7 }$ 
D)  $x = - 7 , x = 7$ 
E)  no zeros

Question 19

Which of the following is the graph of the given equation?
$f ( x ) = \frac { x - 2 } { x - 3 }$

Accepted Answer

A) 
  
B) 
  
C) 
  
D)   
A) 
  
B) 
  
C) 
  
D)

Question 20

Find two positive real numbers whose product is a maximum and whose sum is $146 .$

Accepted Answer

A)  71,75 
B)  73,73 
C)  78,68 
D)  82,64 
E)  61,85 
A)  71,75 
B)  73,73 
C)  78,68 
D)  82,64 
E)  61,85

Determine the equations of any horizontal and vertical asymptotes of $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 16 } .$

Find the zeros of the function below algebraically, if any exist. $f ( x ) = x ^ { 6 } - 9 x ^ { 3 } + 8$

Write the complex conjugate of the complex number $3 - \sqrt { 2 } i$ .

Use long division to divide. $\left( x ^ { 4 } - x ^ { 2 } - 5 \right) \div \left( x ^ { 2 } + 4 x - 1 \right)$

Given $f ( x ) = \frac { 5 x + 4 } { 5 x ^ { 2 } + 4 x }$ . Determine the domain of $f ( x )$ and find any vertical asymptotes.

Simplify $\frac { 1 - 2 i } { 5 i }$ and write the answer in standard form.

Simplify $( 3 - 6 i ) ^ { 2 } - ( 3 + 6 i ) ^ { 2 }$ and write the answer in standard form.

Find two positive real numbers whose product is a maximum and whose sum is 146 .

Determine the zeros (if any) of the rational function $g ( x ) = \frac { x ^ { 3 } - 1 } { x ^ { 2 } + 5 }$ .

Describe the right-hand and the left-hand behavior of the graph of $q ( x ) = 7 x ^ { 5 } + x ^ { 3 } + 7$

Determine the zeros (if any) of the rational function $g ( x ) = 7 + \frac { 2 } { x ^ { 2 } + 7 }$ .

Determine the value that $f ( x ) = \frac { 4 x - 6 } { x ^ { 2 } - 7 }$ approaches as $x$ increases and decreases in magnitude without bound.

Simplify $\frac { 2 + 5 i } { 3 i }$ and write the answer in standard form.

Find all zeros of the function $f ( x ) = ( x + 6 ) ( x + 3 i ) ( x - 3 i )$ .

Simplify $\frac { 4 + 3 i } { 5 + 2 i }$ and write the answer in standard form.

Simplify $( - 4 + i ) ( 4 - 5 i )$ and write the answer in standard form.

Find a fifth degree polynomial function of the lowest degree that has the zeros below and whose leading coefficient is one. $- 3 , - 1,0,1,3$

Determine the zeros (if any) of the rational function $g ( x ) = 7 + \frac { 3 } { x ^ { 2 } + 7 }$ .

Which of the following is the graph of the given equation? $f ( x ) = \frac { x - 2 } { x - 3 }$

Find two positive real numbers whose product is a maximum and whose sum is $146 .$

Functions and Their Graphs

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Filters

Exam 2: Polynomial and Rational Functions

Determine the equations of any horizontal and vertical asymptotes of f(x)=x2x2+16.f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 16 } .f(x)=x2+16x2​.

Find the zeros of the function below algebraically, if any exist. f(x)=x6−9x3+8f ( x ) = x ^ { 6 } - 9 x ^ { 3 } + 8f(x)=x6−9x3+8

Write the complex conjugate of the complex number 3−2i3 - \sqrt { 2 } i3−2​i .

Use long division to divide. (x4−x2−5)÷(x2+4x−1)\left( x ^ { 4 } - x ^ { 2 } - 5 \right) \div \left( x ^ { 2 } + 4 x - 1 \right)(x4−x2−5)÷(x2+4x−1)

Given f(x)=5x+45x2+4xf ( x ) = \frac { 5 x + 4 } { 5 x ^ { 2 } + 4 x }f(x)=5x2+4x5x+4​ . Determine the domain of f(x)f ( x )f(x) and find any vertical asymptotes.

Simplify 1−2i5i\frac { 1 - 2 i } { 5 i }5i1−2i​ and write the answer in standard form.

Simplify (3−6i)2−(3+6i)2( 3 - 6 i ) ^ { 2 } - ( 3 + 6 i ) ^ { 2 }(3−6i)2−(3+6i)2 and write the answer in standard form.

Find two positive real numbers whose product is a maximum and whose sum is 146 .

Determine the zeros (if any) of the rational function g(x)=x3−1x2+5g ( x ) = \frac { x ^ { 3 } - 1 } { x ^ { 2 } + 5 }g(x)=x2+5x3−1​ .

Describe the right-hand and the left-hand behavior of the graph of q(x)=7x5+x3+7q ( x ) = 7 x ^ { 5 } + x ^ { 3 } + 7q(x)=7x5+x3+7

Determine the zeros (if any) of the rational function g(x)=7+2x2+7g ( x ) = 7 + \frac { 2 } { x ^ { 2 } + 7 }g(x)=7+x2+72​ .

Determine the value that f(x)=4x−6x2−7f ( x ) = \frac { 4 x - 6 } { x ^ { 2 } - 7 }f(x)=x2−74x−6​ approaches as xxx increases and decreases in magnitude without bound.

Simplify 2+5i3i\frac { 2 + 5 i } { 3 i }3i2+5i​ and write the answer in standard form.

Find all zeros of the function f(x)=(x+6)(x+3i)(x−3i)f ( x ) = ( x + 6 ) ( x + 3 i ) ( x - 3 i )f(x)=(x+6)(x+3i)(x−3i) .

Simplify 4+3i5+2i\frac { 4 + 3 i } { 5 + 2 i }5+2i4+3i​ and write the answer in standard form.

Simplify (−4+i)(4−5i)( - 4 + i ) ( 4 - 5 i )(−4+i)(4−5i) and write the answer in standard form.

Find a fifth degree polynomial function of the lowest degree that has the zeros below and whose leading coefficient is one. −3,−1,0,1,3- 3 , - 1,0,1,3−3,−1,0,1,3

Determine the zeros (if any) of the rational function g(x)=7+3x2+7g ( x ) = 7 + \frac { 3 } { x ^ { 2 } + 7 }g(x)=7+x2+73​ .

Which of the following is the graph of the given equation? f(x)=x−2x−3f ( x ) = \frac { x - 2 } { x - 3 }f(x)=x−3x−2​

Find two positive real numbers whose product is a maximum and whose sum is 146.146 .146.

Functions and Their Graphs

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Filters

Determine the equations of any horizontal and vertical asymptotes of $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 16 } .$

Find the zeros of the function below algebraically, if any exist. $f ( x ) = x ^ { 6 } - 9 x ^ { 3 } + 8$

Write the complex conjugate of the complex number $3 - \sqrt { 2 } i$ .

Use long division to divide. $\left( x ^ { 4 } - x ^ { 2 } - 5 \right) \div \left( x ^ { 2 } + 4 x - 1 \right)$

Given $f ( x ) = \frac { 5 x + 4 } { 5 x ^ { 2 } + 4 x }$ . Determine the domain of $f ( x )$ and find any vertical asymptotes.

Simplify $\frac { 1 - 2 i } { 5 i }$ and write the answer in standard form.

Simplify $( 3 - 6 i ) ^ { 2 } - ( 3 + 6 i ) ^ { 2 }$ and write the answer in standard form.

Determine the zeros (if any) of the rational function $g ( x ) = \frac { x ^ { 3 } - 1 } { x ^ { 2 } + 5 }$ .

Describe the right-hand and the left-hand behavior of the graph of $q ( x ) = 7 x ^ { 5 } + x ^ { 3 } + 7$

Determine the zeros (if any) of the rational function $g ( x ) = 7 + \frac { 2 } { x ^ { 2 } + 7 }$ .

Determine the value that $f ( x ) = \frac { 4 x - 6 } { x ^ { 2 } - 7 }$ approaches as $x$ increases and decreases in magnitude without bound.

Simplify $\frac { 2 + 5 i } { 3 i }$ and write the answer in standard form.

Find all zeros of the function $f ( x ) = ( x + 6 ) ( x + 3 i ) ( x - 3 i )$ .

Simplify $\frac { 4 + 3 i } { 5 + 2 i }$ and write the answer in standard form.

Simplify $( - 4 + i ) ( 4 - 5 i )$ and write the answer in standard form.

Find a fifth degree polynomial function of the lowest degree that has the zeros below and whose leading coefficient is one. $- 3 , - 1,0,1,3$

Determine the zeros (if any) of the rational function $g ( x ) = 7 + \frac { 3 } { x ^ { 2 } + 7 }$ .

Which of the following is the graph of the given equation? $f ( x ) = \frac { x - 2 } { x - 3 }$

Find two positive real numbers whose product is a maximum and whose sum is $146 .$