Question 1

Given $3 + i$ is a root, determine all other roots of $f ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 42 x ^ { 2 } - 88 x + 80$.

Accepted Answer

A)  $x = 3 + i , 2 \pm 2 i , 2 - i$ 
B)  $x = 3 - i , 2 \pm i$ 
C)  $x = 3 - i , 2 - 2 i , 2 + i$ 
D)  $x = 3 - i , - 2 \pm 2 i$ 
E)  $x = 3 - i , 2 \pm 2 i$ 
A)  $x = 3 + i , 2 \pm 2 i , 2 - i$ 
B)  $x = 3 - i , 2 \pm i$ 
C)  $x = 3 - i , 2 - 2 i , 2 + i$ 
D)  $x = 3 - i , - 2 \pm 2 i$ 
E)  $x = 3 - i , 2 \pm 2 i$

Question 2

Simplify $\frac { - 1 - 5 i } { 7 i }$ and write the answer in standard form.

Accepted Answer

A)  $\frac { 5 } { 7 } + \frac { i } { 7 }$ 
B)  $- \frac { 5 } { 7 } + \frac { i } { 7 }$ 
C)  $- \frac { 5 } { 7 } - \frac { i } { 7 }$ 
D)  $- \frac { 1 } { 7 } - \frac { 5 i } { 7 }$ 
E)  $\frac { 1 } { 7 } - \frac { 5 i } { 7 }$ 
A)  $\frac { 5 } { 7 } + \frac { i } { 7 }$ 
B)  $- \frac { 5 } { 7 } + \frac { i } { 7 }$ 
C)  $- \frac { 5 } { 7 } - \frac { i } { 7 }$ 
D)  $- \frac { 1 } { 7 } - \frac { 5 i } { 7 }$ 
E)  $\frac { 1 } { 7 } - \frac { 5 i } { 7 }$

Question 3

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 4

Determine the domain of $f ( x ) = \frac { 3 x + 3 } { x ^ { 2 } - 3 x }$.

Accepted Answer

A)  all real numbers except $x = - 1 , x = 0$, and $x = 3$ 
B)  all real numbers except $x = 0$ and $x = 3$ 
C)  all real numbers except $x = - 3$ and $x = - 1$ 
D)  all real numbers except $x = 3$ 
E)  all real numbers 
A)  all real numbers except $x = - 1 , x = 0$, and $x = 3$ 
B)  all real numbers except $x = 0$ and $x = 3$ 
C)  all real numbers except $x = - 3$ and $x = - 1$ 
D)  all real numbers except $x = 3$ 
E)  all real numbers

Question 5

Find all the rational zeros of the function $f ( x ) = 3 x ^ { 4 } - 16 x ^ { 3 } - 59 x ^ { 2 } + 400 x - 400$.

Accepted Answer

A)  $x = - 4,5 , - 5 , - \frac { 3 } { 4 }$ 
B)  $x = 3 , - 20,5$ 
C)  $x = 4,5 , - 5 , \frac { 4 } { 3 }$ 
D)  $x = - \frac { 4 } { 5 } , \frac { 5 } { 3 } , \frac { 4 } { 3 } , - 5$ 
E)  $x = 3 , - 20 , \frac { 5 } { 3 } , \frac { 4 } { 3 }$ 
A)  $x = - 4,5 , - 5 , - \frac { 3 } { 4 }$ 
B)  $x = 3 , - 20,5$ 
C)  $x = 4,5 , - 5 , \frac { 4 } { 3 }$ 
D)  $x = - \frac { 4 } { 5 } , \frac { 5 } { 3 } , \frac { 4 } { 3 } , - 5$ 
E)  $x = 3 , - 20 , \frac { 5 } { 3 } , \frac { 4 } { 3 }$

Question 6

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 7

Find the quadratic function $f$ whose graph intersects the $x$-axis at $( 2,0 )$ and $( 3,0 )$ and the $\mathrm { y }$-axis at $( 0 , - 18 )$.

Accepted Answer

A)  $f ( x ) = 3 x ^ { 2 } + 3 x + 9$ 
B)  $f ( x ) = - 3 x ^ { 2 } + 15 x - 18$ 
C)  $f ( x ) = - 3 x ^ { 2 } - 3 x + 6$ 
D)  $f ( x ) = 3 x ^ { 2 } - 3 x - 18$ 
E)  $f ( x ) = 3 x ^ { 2 } - 15 x - 18$ 
A)  $f ( x ) = 3 x ^ { 2 } + 3 x + 9$ 
B)  $f ( x ) = - 3 x ^ { 2 } + 15 x - 18$ 
C)  $f ( x ) = - 3 x ^ { 2 } - 3 x + 6$ 
D)  $f ( x ) = 3 x ^ { 2 } - 3 x - 18$ 
E)  $f ( x ) = 3 x ^ { 2 } - 15 x - 18$

Question 8

Determine the zeros (if any) of the rational function $f ( x ) = \frac { x ^ { 2 } - 64 } { x + 5 }$.

Accepted Answer

A)  $x = - 5$ 
B)  $x = - \frac { 8 } { 5 } , x = \frac { 8 } { 5 }$ 
C)  $x = - 64 , x = 64$ 
D)  $x = - 8 , x = 8$ 
E)  no zeros 
A)  $x = - 5$ 
B)  $x = - \frac { 8 } { 5 } , x = \frac { 8 } { 5 }$ 
C)  $x = - 64 , x = 64$ 
D)  $x = - 8 , x = 8$ 
E)  no zeros

Question 9

Determine the zeros (if any) of the rational function $f ( x ) = \frac { x ^ { 2 } - 9 } { x - 2 }$.

Accepted Answer

A)  $x = 2$ 
B)  $x = \frac { 3 } { 2 } , x = - \frac { 3 } { 2 }$ 
C)  $x = - 9 , x = 9$ 
D)  $x = - 3 , x = 3$ 
E)  no zeros 
A)  $x = 2$ 
B)  $x = \frac { 3 } { 2 } , x = - \frac { 3 } { 2 }$ 
C)  $x = - 9 , x = 9$ 
D)  $x = - 3 , x = 3$ 
E)  no zeros

Question 10

Sketch the graph of the quadratic function below. $h ( x ) = - x ^ { 2 } - 4 x - 1$

Accepted Answer

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

Question 11

Simplify $( \sqrt { - 3 } ) ^ { 9 }$ and write the answer in standard form.

Accepted Answer

A)  $- 81 \sqrt { 3 } i$ 
B)  $81 \sqrt { 3 } i$ 
C)  $6561 \sqrt { 3 } i$ 
D)  $81 \sqrt { 3 }$ 
E)  The expression cannot be simplified. 
A)  $- 81 \sqrt { 3 } i$ 
B)  $81 \sqrt { 3 } i$ 
C)  $6561 \sqrt { 3 } i$ 
D)  $81 \sqrt { 3 }$ 
E)  The expression cannot be simplified.

Question 12

Find all the rational zeros of the function $f ( x ) = 2 x ^ { 4 } - 9 x ^ { 3 } - 41 x ^ { 2 } + 225 x - 225$

Accepted Answer

A) 
$x = - 3,5 , - 5 , - \frac { 2 } { 3 }$ 
B)  $x = 2 , - 15,5$ 
C)  $x = 3,5 , - 5 , \frac { 3 } { 2 }$ 
D)  $x = - \frac { 3 } { 5 } , \frac { 5 } { 2 } , \frac { 3 } { 2 } , - 5$ 
E)  $x = 2 , - 15 , \frac { 5 } { 2 } , \frac { 3 } { 2 }$ 
A) 
$x = - 3,5 , - 5 , - \frac { 2 } { 3 }$ 
B)  $x = 2 , - 15,5$ 
C)  $x = 3,5 , - 5 , \frac { 3 } { 2 }$ 
D)  $x = - \frac { 3 } { 5 } , \frac { 5 } { 2 } , \frac { 3 } { 2 } , - 5$ 
E)  $x = 2 , - 15 , \frac { 5 } { 2 } , \frac { 3 } { 2 }$

Question 13

Find two positive real numbers whose product is a maximum and whose sum is 116 .

Accepted Answer

A)  56,60 
B)  58,58 
C)  63,53 
D)  67,49 
E)  46,70 
A)  56,60 
B)  58,58 
C)  63,53 
D)  67,49 
E)  46,70

Question 14

Determine whether the scatter plot could best be modeled by a linear model, a quadratic model, or neither.

Accepted Answer

A)  linear model 
B)  quadratic model 
C)  neither 
A)  linear model 
B)  quadratic model 
C)  neither

Question 15

Determine the vertex of the graph of the quadratic function $f ( x ) = x ^ { 2 } - x + \frac { 5 } { 4 }$.

Accepted Answer

A)  $\left( \frac { 1 } { 2 } , \frac { 3 } { 2 } \right)$ 
B)  $\left( - 1 , \frac { 5 } { 4 } \right)$ 
C)  $\left( - \frac { 1 } { 2 } , \frac { 5 } { 4 } \right)$ 
D)  $\left( - \frac { 1 } { 4 } , - \frac { 3 } { 4 } \right)$ 
E)  $\left( \frac { 1 } { 2 } , 1 \right)$ 
A)  $\left( \frac { 1 } { 2 } , \frac { 3 } { 2 } \right)$ 
B)  $\left( - 1 , \frac { 5 } { 4 } \right)$ 
C)  $\left( - \frac { 1 } { 2 } , \frac { 5 } { 4 } \right)$ 
D)  $\left( - \frac { 1 } { 4 } , - \frac { 3 } { 4 } \right)$ 
E)  $\left( \frac { 1 } { 2 } , 1 \right)$

Question 16

Sketch the graph of the rational function below. $$f ( x ) = \frac { x ^ { 2 } + 1 } { x }$$

Accepted Answer

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

Question 17

If $x = - 1$ is a root of $x ^ { 3 } + 2 x ^ { 2 } - x - 2 = 0$, use synthetic division to factor the polynomial completely and list all real solutions of the equation.

Accepted Answer

A)  $( x - 2 ) ( x + 1 ) ( x - 1 ) ; x = 2 , - 1,1$ 
B)  $( x + 2 ) ( x + 1 ) ( x - 1 ) ; x = - 2 , - 1,1$ 
C)  $( x + 2 ) ( x + 1 ) ^ { 2 } ; x = - 2 , - 1$ 
D)  $( x + 2 ) ^ { 2 } ( x + 1 ) ; x = - 2 , - 1$ 
E)  $( x + 2 ) ( x - 2 ) ( x - 1 ) ; x = - 2,2,1$ 
A)  $( x - 2 ) ( x + 1 ) ( x - 1 ) ; x = 2 , - 1,1$ 
B)  $( x + 2 ) ( x + 1 ) ( x - 1 ) ; x = - 2 , - 1,1$ 
C)  $( x + 2 ) ( x + 1 ) ^ { 2 } ; x = - 2 , - 1$ 
D)  $( x + 2 ) ^ { 2 } ( x + 1 ) ; x = - 2 , - 1$ 
E)  $( x + 2 ) ( x - 2 ) ( x - 1 ) ; x = - 2,2,1$

Question 18

Use synthetic division to divide.
$$\left( x ^ { 3 } - 75 x + 250 \right) \div ( x - 5 )$$

Accepted Answer

A)  $x ^ { 2 } + 5 x - 50$ 
B)  $x ^ { 2 } - 5 x - 75$ 
C)  $x ^ { 2 } + 10 x + 25$ 
D)  $x ^ { 2 } + 15 x + 50$ 
E)  $x ^ { 2 } + 25 x - 10$ 
A)  $x ^ { 2 } + 5 x - 50$ 
B)  $x ^ { 2 } - 5 x - 75$ 
C)  $x ^ { 2 } + 10 x + 25$ 
D)  $x ^ { 2 } + 15 x + 50$ 
E)  $x ^ { 2 } + 25 x - 10$

Question 19

Describe the right-hand and the left-hand behavior of the graph of $t ( x ) = - \frac { 4 } { 7 } \left( x ^ { 3 } + 5 x ^ { 2 } + 8 x + 1 \right)$.

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 negative, the graph rises to the left and falls to the right. 
C)  Because the degree is odd and the leading coefficient is negative, 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 negative, the graph rises to the left and falls 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 negative, the graph rises to the left and falls to the right. 
C)  Because the degree is odd and the leading coefficient is negative, 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 negative, the graph rises to the left and falls to the right.

Question 20

Identify any horizontal and vertical asymptotes of the function below. $$f ( x ) = \frac { 2 x - 8 } { | x | + 6 }$$

Accepted Answer

A)  vertical asymptotes: $x = - 2$ and $x = 2$; horizontal asymptotes: $y = - 6$ and $y = 6$ 
B)  vertical asymptotes: $x = - 2$ and $x = 2$; horizontal asymptotes: none 
C)  vertical asymptotes: $x = - 6$ and $x = 6$; horizontal asymptotes: none 
D)  vertical asymptotes: none; horizontal asymptotes: $y = - 2$ and $y = 2$ 
E)  vertical asymptotes: $x = - 6$ and $x = 6$; horizontal asymptotes: $y = - 2$ and $y = 2$ 
A)  vertical asymptotes: $x = - 2$ and $x = 2$; horizontal asymptotes: $y = - 6$ and $y = 6$ 
B)  vertical asymptotes: $x = - 2$ and $x = 2$; horizontal asymptotes: none 
C)  vertical asymptotes: $x = - 6$ and $x = 6$; horizontal asymptotes: none 
D)  vertical asymptotes: none; horizontal asymptotes: $y = - 2$ and $y = 2$ 
E)  vertical asymptotes: $x = - 6$ and $x = 6$; horizontal asymptotes: $y = - 2$ and $y = 2$

Given $3 + i$ is a root, determine all other roots of $f ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 42 x ^ { 2 } - 88 x + 80$ .

Simplify $\frac { - 1 - 5 i } { 7 i }$ and write the answer in standard form.

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

Determine the domain of $f ( x ) = \frac { 3 x + 3 } { x ^ { 2 } - 3 x }$ .

Find all the rational zeros of the function $f ( x ) = 3 x ^ { 4 } - 16 x ^ { 3 } - 59 x ^ { 2 } + 400 x - 400$ .

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

Find the quadratic function $f$ whose graph intersects the $x$ -axis at $( 2,0 )$ and $( 3,0 )$ and the $\mathrm { y }$ -axis at $( 0 , - 18 )$ .

Determine the zeros (if any) of the rational function $f ( x ) = \frac { x ^ { 2 } - 64 } { x + 5 }$ .

Determine the zeros (if any) of the rational function $f ( x ) = \frac { x ^ { 2 } - 9 } { x - 2 }$ .

Sketch the graph of the quadratic function below. $h ( x ) = - x ^ { 2 } - 4 x - 1$

Simplify $( \sqrt { - 3 } ) ^ { 9 }$ and write the answer in standard form.

Find all the rational zeros of the function $f ( x ) = 2 x ^ { 4 } - 9 x ^ { 3 } - 41 x ^ { 2 } + 225 x - 225$

Find two positive real numbers whose product is a maximum and whose sum is 116 .

Determine whether the scatter plot could best be modeled by a linear model, a quadratic model, or neither.

Determine the vertex of the graph of the quadratic function $f ( x ) = x ^ { 2 } - x + \frac { 5 } { 4 }$ .

Sketch the graph of the rational function below. $f ( x ) = \frac { x ^ { 2 } + 1 } { x }$

If $x = - 1$ is a root of $x ^ { 3 } + 2 x ^ { 2 } - x - 2 = 0$ , use synthetic division to factor the polynomial completely and list all real solutions of the equation.

Use synthetic division to divide. $\left( x ^ { 3 } - 75 x + 250 \right) \div ( x - 5 )$

Describe the right-hand and the left-hand behavior of the graph of $t ( x ) = - \frac { 4 } { 7 } \left( x ^ { 3 } + 5 x ^ { 2 } + 8 x + 1 \right)$ .

Identify any horizontal and vertical asymptotes of the function below. $f ( x ) = \frac { 2 x - 8 } { | x | + 6 }$

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

Calculus Practice Problems

Filters

Exam 2: Polynomial and Rational Functions

Given 3+i3 + i3+i is a root, determine all other roots of f(x)=x4−10x3+42x2−88x+80f ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 42 x ^ { 2 } - 88 x + 80f(x)=x4−10x3+42x2−88x+80 .

Simplify −1−5i7i\frac { - 1 - 5 i } { 7 i }7i−1−5i​ and write the answer in standard form.

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

Determine the domain of f(x)=3x+3x2−3xf ( x ) = \frac { 3 x + 3 } { x ^ { 2 } - 3 x }f(x)=x2−3x3x+3​ .

Find all the rational zeros of the function f(x)=3x4−16x3−59x2+400x−400f ( x ) = 3 x ^ { 4 } - 16 x ^ { 3 } - 59 x ^ { 2 } + 400 x - 400f(x)=3x4−16x3−59x2+400x−400 .

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 the quadratic function fff whose graph intersects the xxx -axis at (2,0)( 2,0 )(2,0) and (3,0)( 3,0 )(3,0) and the y\mathrm { y }y -axis at (0,−18)( 0 , - 18 )(0,−18) .

Determine the zeros (if any) of the rational function f(x)=x2−64x+5f ( x ) = \frac { x ^ { 2 } - 64 } { x + 5 }f(x)=x+5x2−64​ .

Determine the zeros (if any) of the rational function f(x)=x2−9x−2f ( x ) = \frac { x ^ { 2 } - 9 } { x - 2 }f(x)=x−2x2−9​ .

Sketch the graph of the quadratic function below. h(x)=−x2−4x−1h ( x ) = - x ^ { 2 } - 4 x - 1h(x)=−x2−4x−1

Simplify (−3)9( \sqrt { - 3 } ) ^ { 9 }(−3​)9 and write the answer in standard form.

Find all the rational zeros of the function f(x)=2x4−9x3−41x2+225x−225f ( x ) = 2 x ^ { 4 } - 9 x ^ { 3 } - 41 x ^ { 2 } + 225 x - 225f(x)=2x4−9x3−41x2+225x−225

Find two positive real numbers whose product is a maximum and whose sum is 116 .

Determine whether the scatter plot could best be modeled by a linear model, a quadratic model, or neither.

Determine the vertex of the graph of the quadratic function f(x)=x2−x+54f ( x ) = x ^ { 2 } - x + \frac { 5 } { 4 }f(x)=x2−x+45​ .

Sketch the graph of the rational function below. f(x)=x2+1xf ( x ) = \frac { x ^ { 2 } + 1 } { x }f(x)=xx2+1​

If x=−1x = - 1x=−1 is a root of x3+2x2−x−2=0x ^ { 3 } + 2 x ^ { 2 } - x - 2 = 0x3+2x2−x−2=0 , use synthetic division to factor the polynomial completely and list all real solutions of the equation.

Use synthetic division to divide. (x3−75x+250)÷(x−5)\left( x ^ { 3 } - 75 x + 250 \right) \div ( x - 5 )(x3−75x+250)÷(x−5)

Describe the right-hand and the left-hand behavior of the graph of t(x)=−47(x3+5x2+8x+1)t ( x ) = - \frac { 4 } { 7 } \left( x ^ { 3 } + 5 x ^ { 2 } + 8 x + 1 \right)t(x)=−74​(x3+5x2+8x+1) .

Identify any horizontal and vertical asymptotes of the function below. f(x)=2x−8∣x∣+6f ( x ) = \frac { 2 x - 8 } { | x | + 6 }f(x)=∣x∣+62x−8​

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

Calculus Practice Problems

Filters

Given $3 + i$ is a root, determine all other roots of $f ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 42 x ^ { 2 } - 88 x + 80$ .

Simplify $\frac { - 1 - 5 i } { 7 i }$ and write the answer in standard form.

Determine the domain of $f ( x ) = \frac { 3 x + 3 } { x ^ { 2 } - 3 x }$ .

Find all the rational zeros of the function $f ( x ) = 3 x ^ { 4 } - 16 x ^ { 3 } - 59 x ^ { 2 } + 400 x - 400$ .

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

Find the quadratic function $f$ whose graph intersects the $x$ -axis at $( 2,0 )$ and $( 3,0 )$ and the $\mathrm { y }$ -axis at $( 0 , - 18 )$ .

Determine the zeros (if any) of the rational function $f ( x ) = \frac { x ^ { 2 } - 64 } { x + 5 }$ .

Determine the zeros (if any) of the rational function $f ( x ) = \frac { x ^ { 2 } - 9 } { x - 2 }$ .

Sketch the graph of the quadratic function below. $h ( x ) = - x ^ { 2 } - 4 x - 1$

Simplify $( \sqrt { - 3 } ) ^ { 9 }$ and write the answer in standard form.

Find all the rational zeros of the function $f ( x ) = 2 x ^ { 4 } - 9 x ^ { 3 } - 41 x ^ { 2 } + 225 x - 225$

Determine the vertex of the graph of the quadratic function $f ( x ) = x ^ { 2 } - x + \frac { 5 } { 4 }$ .

Sketch the graph of the rational function below. $f ( x ) = \frac { x ^ { 2 } + 1 } { x }$

If $x = - 1$ is a root of $x ^ { 3 } + 2 x ^ { 2 } - x - 2 = 0$ , use synthetic division to factor the polynomial completely and list all real solutions of the equation.

Use synthetic division to divide. $\left( x ^ { 3 } - 75 x + 250 \right) \div ( x - 5 )$

Describe the right-hand and the left-hand behavior of the graph of $t ( x ) = - \frac { 4 } { 7 } \left( x ^ { 3 } + 5 x ^ { 2 } + 8 x + 1 \right)$ .

Identify any horizontal and vertical asymptotes of the function below. $f ( x ) = \frac { 2 x - 8 } { | x | + 6 }$