Question 1

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 2

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 3

Use long division to divide.
$\left( x ^ { 3 } + 3 x ^ { 2 } + x + 3 \right) \div ( x + 3 )$

Accepted Answer

A)  $x ^ { 2 } + 3$ 
B)  $x ^ { 2 } + 6 x + 17 - \frac { 53 } { x + 3 }$ 
C)  $x ^ { 2 } + 6 x + 19 + \frac { 48 } { x + 3 }$ 
D)  $x ^ { 2 } + 6 x + 17$ 
E)  $x ^ { 2 } + 1$ 
A)  $x ^ { 2 } + 3$ 
B)  $x ^ { 2 } + 6 x + 17 - \frac { 53 } { x + 3 }$ 
C)  $x ^ { 2 } + 6 x + 19 + \frac { 48 } { x + 3 }$ 
D)  $x ^ { 2 } + 6 x + 17$ 
E)  $x ^ { 2 } + 1$

Question 4

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 5

The graph of the function
$f ( x ) = \frac { 3 x } { x + 1 }$ is shown below. Determine the domain.

Accepted Answer

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

Question 6

Determine the domain of $f ( x ) = \frac { 6 x + 6 } { x ^ { 2 } - 6 x }$.

Accepted Answer

A)  all real numbers except $x = - 1 , x = 0$, and $x = 6$ 
B)  all real numbers except $x = 0$ and $x = 6$ 
C)  all real numbers except $x = - 6$ and $x = - 1$ 
D)  all real numbers except $x = 6$ 
E)  all real numbers 
A)  all real numbers except $x = - 1 , x = 0$, and $x = 6$ 
B)  all real numbers except $x = 0$ and $x = 6$ 
C)  all real numbers except $x = - 6$ and $x = - 1$ 
D)  all real numbers except $x = 6$ 
E)  all real numbers

Question 7

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 8

Perform the following operation and write the result in standard form.
$$\frac { 9 i } { 9 + i } + \frac { 2 } { 9 - i }$$

Accepted Answer

A)  $\frac { 9 } { 40 } + \frac { 83 } { 80 } i$ 
B)  $\frac { 27 } { 10 } + \frac { 83 } { 10 } i$ 
C)  $\frac { 1 } { 41 } + \frac { 9 } { 82 } i$ 
D)  $\frac { 27 } { 82 } + \frac { 83 } { 82 } i$ 
E)  $\frac { 27 } { 8 } + \frac { 83 } { 8 } i$ 
A)  $\frac { 9 } { 40 } + \frac { 83 } { 80 } i$ 
B)  $\frac { 27 } { 10 } + \frac { 83 } { 10 } i$ 
C)  $\frac { 1 } { 41 } + \frac { 9 } { 82 } i$ 
D)  $\frac { 27 } { 82 } + \frac { 83 } { 82 } i$ 
E)  $\frac { 27 } { 8 } + \frac { 83 } { 8 } i$

Question 9

Determine the vertex of the graph of the quadratic function 
$f ( x ) = x ^ { 2 } + 5 x + \frac { 29 } { 4 }$.

Accepted Answer

A)  $\left( \frac { - 5 } { 2 } , \frac { 27 } { 2 } \right)$ 
B)  $\left( 5 , \frac { 29 } { 4 } \right)$ 
C)  $\left( \frac { 5 } { 2 } , \frac { 29 } { 4 } \right)$ 
D)  $\left( \frac { 5 } { 4 } , \frac { 21 } { 4 } \right)$ 
E)  $\left( \frac { - 5 } { 2 } , 1 \right)$ 
A)  $\left( \frac { - 5 } { 2 } , \frac { 27 } { 2 } \right)$ 
B)  $\left( 5 , \frac { 29 } { 4 } \right)$ 
C)  $\left( \frac { 5 } { 2 } , \frac { 29 } { 4 } \right)$ 
D)  $\left( \frac { 5 } { 4 } , \frac { 21 } { 4 } \right)$ 
E)  $\left( \frac { - 5 } { 2 } , 1 \right)$

Question 10

Use synthetic division to divide.
$$\left( 2 x ^ { 3 } - 5 x ^ { 2 } - 22 x - 15 \right) \div ( x - 5 )$$

Accepted Answer

A)  $2 x ^ { 2 } - 3 x - 5$ 
B)  $2 x ^ { 2 } + 5 x + 3$ 
C)  $2 x ^ { 2 } - 2 x - 15$ 
D)  $2 x ^ { 2 } - 7 x + 6$ 
E)  $2 x ^ { 2 } + 5 x + 2$ 
A)  $2 x ^ { 2 } - 3 x - 5$ 
B)  $2 x ^ { 2 } + 5 x + 3$ 
C)  $2 x ^ { 2 } - 2 x - 15$ 
D)  $2 x ^ { 2 } - 7 x + 6$ 
E)  $2 x ^ { 2 } + 5 x + 2$

Question 11

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 12

A polynomial function $f$ has degree 3 , the zeros below, and a solution point of $f ( - 3 ) = - 4$. Write $f$ in completely factored form. $- 4 , - 3 + 2 i$

Accepted Answer

A)  $f ( x ) = ( x + 3 ) ( x + 4 - 2 i ) ( x + 4 + 2 i )$ 
B)  $f ( x ) = - ( x + 4 ) ( x + 2 - 3 i ) ( x + 2 + 3 i )$ 
C)  $f ( x ) = ( x + 4 ) ( x + 2 - 3 i ) ( x + 2 + 3 i )$ 
D)  $f ( x ) = - ( x + 4 ) ( x + 3 - 2 i ) ( x + 3 + 2 i )$ 
E)  $f ( x ) = ( x + 4 ) ( x + 3 - 2 i ) ( x + 3 + 2 i )$ 
A)  $f ( x ) = ( x + 3 ) ( x + 4 - 2 i ) ( x + 4 + 2 i )$ 
B)  $f ( x ) = - ( x + 4 ) ( x + 2 - 3 i ) ( x + 2 + 3 i )$ 
C)  $f ( x ) = ( x + 4 ) ( x + 2 - 3 i ) ( x + 2 + 3 i )$ 
D)  $f ( x ) = - ( x + 4 ) ( x + 3 - 2 i ) ( x + 3 + 2 i )$ 
E)  $f ( x ) = ( x + 4 ) ( x + 3 - 2 i ) ( x + 3 + 2 i )$

Question 13

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 14

The interest rates that banks charge to borrow money fluctuate with the economy. The interest rate charged by a bank in a certain country is given in the table below. Let $t$ represent the year, with $t = 0$ corresponding to 1986 . Use the regression feature of a graphing utility to find a quadratic model of the form $y = a t ^ { 2 } + b t + c$ for the data.
$\begin{array}{|c|c|}
\hline \begin{array}{c}
\text { Year } \
\boldsymbol{t}
\end{array} & \begin{array}{c}
\text { Percent } \
\boldsymbol{y}
\end{array} \
\hline 1986 & 12.8 \
\hline 1988 & 10.0 \
\hline 1990 & 6.9 \
\hline 1992 & 5.7 \
\hline 1994 & 8.6 \
\hline 1996 & 12.7 \
\hline
\end{array}$

Accepted Answer

A)  $y = - 2.7 t ^ { 2 } + 13.35 t + 0.26$ 
B)  $y = 13.35 t ^ { 2 } + 0.26 t - 2.7$ 
C)  $y = 0.26 t ^ { 2 } - 2.7 t + 13.35$ 
D)  $y = 0.21 t ^ { 2 } - 3.26 t + 11.22$ 
E)  $y = 0.32 t ^ { 2 } - 2.18 t + 15.22$ 
A)  $y = - 2.7 t ^ { 2 } + 13.35 t + 0.26$ 
B)  $y = 13.35 t ^ { 2 } + 0.26 t - 2.7$ 
C)  $y = 0.26 t ^ { 2 } - 2.7 t + 13.35$ 
D)  $y = 0.21 t ^ { 2 } - 3.26 t + 11.22$ 
E)  $y = 0.32 t ^ { 2 } - 2.18 t + 15.22$

Question 15

Use long division to divide.
$$\left( x ^ { 3 } + 27 \right) \div ( x + 3 )$$

Accepted Answer

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

Question 16

Use long division to divide.
$$\left( x ^ { 3 } + 4 x - 1 \right) \div ( x + 2 )$$

Accepted Answer

A)  $x ^ { 2 } - 2 x + 8 - \frac { 17 } { x + 2 }$ 
B)  $x ^ { 2 } + 2 x + 8 - \frac { 15 } { x + 2 }$ 
C)  $x ^ { 2 } + 2 - \frac { 3 } { x + 2 }$ 
D)  $x ^ { 2 } - 2 + \frac { 3 } { x + 2 }$ 
E)  $x ^ { 2 } + 2 x - 8 + \frac { 17 } { x + 2 }$ 
A)  $x ^ { 2 } - 2 x + 8 - \frac { 17 } { x + 2 }$ 
B)  $x ^ { 2 } + 2 x + 8 - \frac { 15 } { x + 2 }$ 
C)  $x ^ { 2 } + 2 - \frac { 3 } { x + 2 }$ 
D)  $x ^ { 2 } - 2 + \frac { 3 } { x + 2 }$ 
E)  $x ^ { 2 } + 2 x - 8 + \frac { 17 } { x + 2 }$

Question 17

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 18

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

Accepted Answer

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

Question 19

Write $f ( x ) = x ^ { 3 } - 11 x ^ { 2 } + 18 x + 25$ in the form $f ( x ) = ( x - k ) q ( x ) + r$ when $k = 6 + \sqrt { 6 }$.

Accepted Answer

A)  $f ( x ) = [ x + ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } + ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] - 5$ 
B)  $f ( x ) = [ x + ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } + ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] + 5$ 
C)  $f ( x ) = [ x - ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } + ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] + 5$ 
D)  $f ( x ) = [ x - ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } + ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] - 5$ 
E)  $f ( x ) = [ x + ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } - ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] - 5$ 
A)  $f ( x ) = [ x + ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } + ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] - 5$ 
B)  $f ( x ) = [ x + ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } + ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] + 5$ 
C)  $f ( x ) = [ x - ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } + ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] + 5$ 
D)  $f ( x ) = [ x - ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } + ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] - 5$ 
E)  $f ( x ) = [ x + ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } - ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] - 5$

Question 20

Find all zeros of the function $f ( x ) = x ^ { 2 } ( x - 2 ) \left( x ^ { 3 } - 216 \right)$.

Accepted Answer

A)  $x = 2,216$ 
B)  $ x = 0 , - 2 , - 6$ 
C)  $ x = 0,2,6 , - 3 - 3 \sqrt { 3 } i , - 3 + 3 \sqrt { 3 } i$ 
D)  $x = - 2 , - 216$ 
E)  $ x = 0,2,6$ 
A)  $x = 2,216$ 
B)  $ x = 0 , - 2 , - 6$ 
C)  $ x = 0,2,6 , - 3 - 3 \sqrt { 3 } i , - 3 + 3 \sqrt { 3 } i$ 
D)  $x = - 2 , - 216$ 
E)  $ x = 0,2,6$

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

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

Use long division to divide. $\left( x ^ { 3 } + 3 x ^ { 2 } + x + 3 \right) \div ( x + 3 )$

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

The graph of the function $f ( x ) = \frac { 3 x } { x + 1 }$ is shown below. Determine the domain. $The graph of the function f ( x ) = \frac { 3 x } { x + 1 } is shown below. Determine the domain.$

Determine the domain of $f ( x ) = \frac { 6 x + 6 } { x ^ { 2 } - 6 x }$ .

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

Perform the following operation and write the result in standard form. $\frac { 9 i } { 9 + i } + \frac { 2 } { 9 - i }$

Determine the vertex of the graph of the quadratic function $f ( x ) = x ^ { 2 } + 5 x + \frac { 29 } { 4 }$ .

Use synthetic division to divide. $\left( 2 x ^ { 3 } - 5 x ^ { 2 } - 22 x - 15 \right) \div ( x - 5 )$

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

A polynomial function $f$ has degree 3 , the zeros below, and a solution point of $f ( - 3 ) = - 4$ . Write $f$ in completely factored form. $- 4 , - 3 + 2 i$

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

Use long division to divide. $\left( x ^ { 3 } + 27 \right) \div ( x + 3 )$

Use long division to divide. $\left( x ^ { 3 } + 4 x - 1 \right) \div ( x + 2 )$

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

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

Write $f ( x ) = x ^ { 3 } - 11 x ^ { 2 } + 18 x + 25$ in the form $f ( x ) = ( x - k ) q ( x ) + r$ when $k = 6 + \sqrt { 6 }$ .

Find all zeros of the function $f ( x ) = x ^ { 2 } ( x - 2 ) \left( x ^ { 3 } - 216 \right)$ .

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

Review of Graphs, Equations, and Inequalities

Filters

Exam 2: Polynomial and Rational Functions

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

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 .

Use long division to divide. (x3+3x2+x+3)÷(x+3)\left( x ^ { 3 } + 3 x ^ { 2 } + x + 3 \right) \div ( x + 3 )(x3+3x2+x+3)÷(x+3)

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

The graph of the function f(x)=3xx+1f ( x ) = \frac { 3 x } { x + 1 }f(x)=x+13x​ is shown below. Determine the domain.

Determine the domain of f(x)=6x+6x2−6xf ( x ) = \frac { 6 x + 6 } { x ^ { 2 } - 6 x }f(x)=x2−6x6x+6​ .

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​ .

Perform the following operation and write the result in standard form. 9i9+i+29−i\frac { 9 i } { 9 + i } + \frac { 2 } { 9 - i }9+i9i​+9−i2​

Determine the vertex of the graph of the quadratic function f(x)=x2+5x+294f ( x ) = x ^ { 2 } + 5 x + \frac { 29 } { 4 }f(x)=x2+5x+429​ .

Use synthetic division to divide. (2x3−5x2−22x−15)÷(x−5)\left( 2 x ^ { 3 } - 5 x ^ { 2 } - 22 x - 15 \right) \div ( x - 5 )(2x3−5x2−22x−15)÷(x−5)

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 .

A polynomial function fff has degree 3 , the zeros below, and a solution point of f(−3)=−4f ( - 3 ) = - 4f(−3)=−4 . Write fff in completely factored form. −4,−3+2i- 4 , - 3 + 2 i−4,−3+2i

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​ .

Use long division to divide. (x3+27)÷(x+3)\left( x ^ { 3 } + 27 \right) \div ( x + 3 )(x3+27)÷(x+3)

Use long division to divide. (x3+4x−1)÷(x+2)\left( x ^ { 3 } + 4 x - 1 \right) \div ( x + 2 )(x3+4x−1)÷(x+2)

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

Which of the following is the graph of the given equation? f(x)=2−xx+3f ( x ) = \frac { 2 - x } { x + 3 }f(x)=x+32−x​

Write f(x)=x3−11x2+18x+25f ( x ) = x ^ { 3 } - 11 x ^ { 2 } + 18 x + 25f(x)=x3−11x2+18x+25 in the form f(x)=(x−k)q(x)+rf ( x ) = ( x - k ) q ( x ) + rf(x)=(x−k)q(x)+r when k=6+6k = 6 + \sqrt { 6 }k=6+6​ .

Find all zeros of the function f(x)=x2(x−2)(x3−216)f ( x ) = x ^ { 2 } ( x - 2 ) \left( x ^ { 3 } - 216 \right)f(x)=x2(x−2)(x3−216) .

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

Review of Graphs, Equations, and Inequalities

Filters

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

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

Use long division to divide. $\left( x ^ { 3 } + 3 x ^ { 2 } + x + 3 \right) \div ( x + 3 )$

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

The graph of the function $f ( x ) = \frac { 3 x } { x + 1 }$ is shown below. Determine the domain. $The graph of the function f ( x ) = \frac { 3 x } { x + 1 } is shown below. Determine the domain.$

Determine the domain of $f ( x ) = \frac { 6 x + 6 } { x ^ { 2 } - 6 x }$ .

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

Perform the following operation and write the result in standard form. $\frac { 9 i } { 9 + i } + \frac { 2 } { 9 - i }$

Determine the vertex of the graph of the quadratic function $f ( x ) = x ^ { 2 } + 5 x + \frac { 29 } { 4 }$ .

Use synthetic division to divide. $\left( 2 x ^ { 3 } - 5 x ^ { 2 } - 22 x - 15 \right) \div ( x - 5 )$

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

A polynomial function $f$ has degree 3 , the zeros below, and a solution point of $f ( - 3 ) = - 4$ . Write $f$ in completely factored form. $- 4 , - 3 + 2 i$

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

Use long division to divide. $\left( x ^ { 3 } + 27 \right) \div ( x + 3 )$

Use long division to divide. $\left( x ^ { 3 } + 4 x - 1 \right) \div ( x + 2 )$

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

Write $f ( x ) = x ^ { 3 } - 11 x ^ { 2 } + 18 x + 25$ in the form $f ( x ) = ( x - k ) q ( x ) + r$ when $k = 6 + \sqrt { 6 }$ .

Find all zeros of the function $f ( x ) = x ^ { 2 } ( x - 2 ) \left( x ^ { 3 } - 216 \right)$ .