Question 1

Determine the value that $f ( x ) = \frac { 3 x - 5 } { x - 6 }$ approaches as $x$ increases and decreases in magnitude without bound.

Accepted Answer

A)  6 
B)  5 
C)  4 
D)  3 
E)  2 
A)  6 
B)  5 
C)  4 
D)  3 
E)  2 D

Question 2

Find the zeros of the function below algebraically, if any exist.
$$f ( t ) = \frac { 1 } { 6 } t ^ { 4 } - \frac { 27 } { 2 }$$

Accepted Answer

A)  $- 3 , - 1,1$, and 3 
B)  $- 3$ and 3 
C)  $- 6$ and 6 
D)  $- 6 , - 1,1$, and 6 
E)  No zeros exist. 
A)  $- 3 , - 1,1$, and 3 
B)  $- 3$ and 3 
C)  $- 6$ and 6 
D)  $- 6 , - 1,1$, and 6 
E)  No zeros exist. B

Question 3

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 A

Question 4

Use the regression feature of a graphing utility to find a quadratic model for the data below. 
$\begin{array}{|r|c|}
\hline x & y \
\hline-2 & -7.3 \
\hline-1 & -2 \
\hline 0 & -1 \
\hline 1 & -4.8 \
\hline 2 & -11.4 \
\hline 3 & -22.8 \
\hline 4 & -37.8 \
\hline
\end{array}$

Accepted Answer

A)  $y = - 1.91 x ^ { 2 } - 1.21 x - 1.57$ 
B)  $y = - 1.81 x ^ { 2 } - 1.15 x - 1.42$ 
C)  $y = - 2.21 x ^ { 2 } - 1.04 x - 1$ 
D)  $y = - 2.11 x ^ { 2 } - 0.99 x - 1.07$ 
E)  $y = - 2.01 x ^ { 2 } - 1.1 x - 1.28$ 
A)  $y = - 1.91 x ^ { 2 } - 1.21 x - 1.57$ 
B)  $y = - 1.81 x ^ { 2 } - 1.15 x - 1.42$ 
C)  $y = - 2.21 x ^ { 2 } - 1.04 x - 1$ 
D)  $y = - 2.11 x ^ { 2 } - 0.99 x - 1.07$ 
E)  $y = - 2.01 x ^ { 2 } - 1.1 x - 1.28$

Question 5

Simplify $f$ below and find any vertical asymptotes of $f$.
$$f ( x ) = \frac { x ^ { 2 } - 25 } { x + 5 }$$

Accepted Answer

A)  $f ( x ) = x + 5 , x \neq 5$; vertical asymptotes: none 
B)  $f ( x ) = x - 5 , x \neq - 5$; vertical asymptotes: none 
C)  $f ( x ) = x - 5 , x \neq 5$; vertical asymptotes: none 
D)  $f ( x ) = x + 5 , x \neq - 5$; vertical asymptotes: $x = - 5$ 
E)  $f ( x ) = x - 5 , x \neq 5$; vertical asymptotes: $x = 5$ 
A)  $f ( x ) = x + 5 , x \neq 5$; vertical asymptotes: none 
B)  $f ( x ) = x - 5 , x \neq - 5$; vertical asymptotes: none 
C)  $f ( x ) = x - 5 , x \neq 5$; vertical asymptotes: none 
D)  $f ( x ) = x + 5 , x \neq - 5$; vertical asymptotes: $x = - 5$ 
E)  $f ( x ) = x - 5 , x \neq 5$; vertical asymptotes: $x = 5$

Question 6

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 7

Find two positive real numbers whose product is a maximum and whose sum of the first number and four times the second is 200 .

Accepted Answer

A)  160,10 
B)  116,21 
C)  108,23 
D)  100,25 
E)  76,31 
A)  160,10 
B)  116,21 
C)  108,23 
D)  100,25 
E)  76,31

Question 8

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

Accepted Answer

A)  $- \frac { 12 } { 29 } + \frac { 1 } { 29 } i$ 
B)  $\frac { 12 } { 29 } - \frac { 1 } { 29 } i$ 
C)  $\frac { 12 } { 29 } + \frac { 1 } { 29 } i$ 
D)  $\frac { 1 } { 29 } + \frac { 12 } { 29 } i$ 
E)  $\frac { 1 } { 29 } - \frac { 12 } { 29 } i$ 
A)  $- \frac { 12 } { 29 } + \frac { 1 } { 29 } i$ 
B)  $\frac { 12 } { 29 } - \frac { 1 } { 29 } i$ 
C)  $\frac { 12 } { 29 } + \frac { 1 } { 29 } i$ 
D)  $\frac { 1 } { 29 } + \frac { 12 } { 29 } i$ 
E)  $\frac { 1 } { 29 } - \frac { 12 } { 29 } i$

Question 9

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 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 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 12

Find all the rational zeros of the function $f ( x ) = - 2 x ^ { 5 } - 11 x ^ { 4 } - 19 x ^ { 3 } - 17 x ^ { 2 } - 17 x - 6$.

Accepted Answer

A)  $x = \frac { 1 } { 2 } , - 3 , - 1$ 
B)  $x = - \frac { 2 } { 3 } , 1 , - 2$ 
C)  $x = - \frac { 1 } { 2 } , \frac { 3 } { 2 } , - 2$ 
D)  $x = - \frac { 1 } { 2 } , \frac { 3 } { 2 }$ 
E)  $\quad x = - \frac { 1 } { 2 } , - 3 , - 2$ 
A)  $x = \frac { 1 } { 2 } , - 3 , - 1$ 
B)  $x = - \frac { 2 } { 3 } , 1 , - 2$ 
C)  $x = - \frac { 1 } { 2 } , \frac { 3 } { 2 } , - 2$ 
D)  $x = - \frac { 1 } { 2 } , \frac { 3 } { 2 }$ 
E)  $\quad x = - \frac { 1 } { 2 } , - 3 , - 2$

Question 13

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 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.
$\begin{array}{|c|c|}
\hline \text { Year } & \text { Percent } \
\boldsymbol{t} & \boldsymbol{y} \
\hline 1986 & 12.8 \
\hline 1988 & 10.0 \
\hline 1990 & 6.9 \
\hline 1992 & 5.7 \
\hline 1994 & 8.6 \
\hline 1996 & 12.7 \
\hline
\end{array}$

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 the regression feature of a graphing utility to find a quadratic model for the data below.
$\begin{array}{|r|c|}
\hline \boldsymbol{x} & \boldsymbol{y} \
\hline-2 & 15.7 \
\hline-1 & 3.5 \
\hline 0 & -2.7 \
\hline 1 & -2.9 \
\hline 2 & 3.7 \
\hline 3 & 15.8 \
\hline 4 & 33.6 \
\hline
\end{array}$

Accepted Answer

A)  $y = 2.89 x ^ { 2 } - 3.35 x - 2.9$ 
B)  $y = 2.73 x ^ { 2 } - 3.2 x - 2.75$ 
C)  $y = 3.34 x ^ { 2 } - 2.9 x - 2.7$ 
D)  $y = 3.19 x ^ { 2 } - 2.74 x - 2.4$ 
E)  $y = 3.04 x ^ { 2 } - 3.05 x - 2.61$ 
A)  $y = 2.89 x ^ { 2 } - 3.35 x - 2.9$ 
B)  $y = 2.73 x ^ { 2 } - 3.2 x - 2.75$ 
C)  $y = 3.34 x ^ { 2 } - 2.9 x - 2.7$ 
D)  $y = 3.19 x ^ { 2 } - 2.74 x - 2.4$ 
E)  $y = 3.04 x ^ { 2 } - 3.05 x - 2.61$

Question 16

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

Accepted Answer

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

Question 17

Using the factors $( - 5 x + 2 )$ and $( x - 1 )$, find the remaining factor(s) of $f ( x ) = 10 x ^ { 4 } + 31 x ^ { 3 } - 84 x ^ { 2 } + 53 x - 10$ and write the polynomial in fully factored form.

Accepted Answer

A)  $f ( x ) = ( - 5 x + 2 ) ( - 5 x + 2 ) ( 2 x - 1 ) ( x - 1 )$ 
B)  $f ( x ) = ( - 5 x + 2 ) ( - x - 5 ) ( 2 x - 1 ) ( x - 1 )$ 
C)  $f ( x ) = ( - 5 x + 2 ) ^ { 2 } ( 2 x - 1 ) ( x + 1 )$ 
D)  $f ( x ) = ( - 5 x + 2 ) ( - x + 5 ) ^ { 2 } ( x + 1 )$ 
E)  $f ( x ) = ( - 5 x + 2 ) ^ { 2 } ( x - 1 ) ^ { 2 }$ 
A)  $f ( x ) = ( - 5 x + 2 ) ( - 5 x + 2 ) ( 2 x - 1 ) ( x - 1 )$ 
B)  $f ( x ) = ( - 5 x + 2 ) ( - x - 5 ) ( 2 x - 1 ) ( x - 1 )$ 
C)  $f ( x ) = ( - 5 x + 2 ) ^ { 2 } ( 2 x - 1 ) ( x + 1 )$ 
D)  $f ( x ) = ( - 5 x + 2 ) ( - x + 5 ) ^ { 2 } ( x + 1 )$ 
E)  $f ( x ) = ( - 5 x + 2 ) ^ { 2 } ( x - 1 ) ^ { 2 }$

Question 18

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.
$\begin{array}{|c|c|}
\hline \text { Year } & \text { Percent } \
\boldsymbol{t} & \boldsymbol{y} \
\hline 1986 & 12.1 \
\hline 1988 & 10.1 \
\hline 1990 & 6.8 \
\hline 1992 & 6.6 \
\hline 1994 & 8.6 \
\hline 1996 & 12.0 \
\hline
\end{array}$

Accepted Answer

A)  $y = - 2.23 t ^ { 2 } + 12.61 t + 0.22$ 
B)  $y = 12.61 t ^ { 2 } + 0.22 t - 2.23$ 
C)  $y = 0.22 t ^ { 2 } - 2.23 t + 12.61$ 
D)  $y = 0.17 t ^ { 2 } - 2.69 t + 10.59$ 
E)  $y = 0.26 t ^ { 2 } - 1.8 t + 14.37$ 
A)  $y = - 2.23 t ^ { 2 } + 12.61 t + 0.22$ 
B)  $y = 12.61 t ^ { 2 } + 0.22 t - 2.23$ 
C)  $y = 0.22 t ^ { 2 } - 2.23 t + 12.61$ 
D)  $y = 0.17 t ^ { 2 } - 2.69 t + 10.59$ 
E)  $y = 0.26 t ^ { 2 } - 1.8 t + 14.37$

Question 19

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 20

Find the zeros of the function below algebraically, if any exist. $$f ( x ) = x ^ { 5 } - 5 x ^ { 3 } + 4 x$$

Accepted Answer

A)  $- 4 , - 1,1,4$ 
B)  $- 4 , - 2,2,4$ 
C)  $- 2 , - 1,0,1,2$ 
D)  $- 4 , - 2,0,2,4$ 
E)  No zeros exist. 
A)  $- 4 , - 1,1,4$ 
B)  $- 4 , - 2,2,4$ 
C)  $- 2 , - 1,0,1,2$ 
D)  $- 4 , - 2,0,2,4$ 
E)  No zeros exist.

Determine the value that $f ( x ) = \frac { 3 x - 5 } { x - 6 }$ approaches as $x$ increases and decreases in magnitude without bound.

Find the zeros of the function below algebraically, if any exist. $f ( t ) = \frac { 1 } { 6 } t ^ { 4 } - \frac { 27 } { 2 }$

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

Use the regression feature of a graphing utility to find a quadratic model for the data below. x y -2 -7.3 -1 -2 0 -1 1 -4.8 2 -11.4 3 -22.8 4 -37.8

Simplify $f$ below and find any vertical asymptotes of $f$ . $f ( x ) = \frac { x ^ { 2 } - 25 } { x + 5 }$

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

Find two positive real numbers whose product is a maximum and whose sum of the first number and four times the second is 200 .

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

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

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

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

Find all the rational zeros of the function $f ( x ) = - 2 x ^ { 5 } - 11 x ^ { 4 } - 19 x ^ { 3 } - 17 x ^ { 2 } - 17 x - 6$ .

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

Use the regression feature of a graphing utility to find a quadratic model for the data below. -2 15.7 -1 3.5 0 -2.7 1 -2.9 2 3.7 3 15.8 4 33.6

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

Using the factors $( - 5 x + 2 )$ and $( x - 1 )$ , find the remaining factor(s) of $f ( x ) = 10 x ^ { 4 } + 31 x ^ { 3 } - 84 x ^ { 2 } + 53 x - 10$ and write the polynomial in fully factored form.

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

Find the zeros of the function below algebraically, if any exist. $f ( x ) = x ^ { 5 } - 5 x ^ { 3 } + 4 x$

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

Determine the value that f(x)=3x−5x−6f ( x ) = \frac { 3 x - 5 } { x - 6 }f(x)=x−63x−5​ approaches as xxx increases and decreases in magnitude without bound.

Find the zeros of the function below algebraically, if any exist. f(t)=16t4−272f ( t ) = \frac { 1 } { 6 } t ^ { 4 } - \frac { 27 } { 2 }f(t)=61​t4−227​

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

Use the regression feature of a graphing utility to find a quadratic model for the data below. x y -2 -7.3 -1 -2 0 -1 1 -4.8 2 -11.4 3 -22.8 4 -37.8

Simplify fff below and find any vertical asymptotes of fff . f(x)=x2−25x+5f ( x ) = \frac { x ^ { 2 } - 25 } { x + 5 }f(x)=x+5x2−25​

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​ .

Find two positive real numbers whose product is a maximum and whose sum of the first number and four times the second is 200 .

Simplify 2+i5+2i\frac { 2 + i } { 5 + 2 i }5+2i2+i​ and write the answer in standard form.

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

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

Find all the rational zeros of the function f(x)=−2x5−11x4−19x3−17x2−17x−6f ( x ) = - 2 x ^ { 5 } - 11 x ^ { 4 } - 19 x ^ { 3 } - 17 x ^ { 2 } - 17 x - 6f(x)=−2x5−11x4−19x3−17x2−17x−6 .

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

Use the regression feature of a graphing utility to find a quadratic model for the data below. -2 15.7 -1 3.5 0 -2.7 1 -2.9 2 3.7 3 15.8 4 33.6

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​

Using the factors (−5x+2)( - 5 x + 2 )(−5x+2) and (x−1)( x - 1 )(x−1) , find the remaining factor(s) of f(x)=10x4+31x3−84x2+53x−10f ( x ) = 10 x ^ { 4 } + 31 x ^ { 3 } - 84 x ^ { 2 } + 53 x - 10f(x)=10x4+31x3−84x2+53x−10 and write the polynomial in fully factored form.

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

Find the zeros of the function below algebraically, if any exist. f(x)=x5−5x3+4xf ( x ) = x ^ { 5 } - 5 x ^ { 3 } + 4 xf(x)=x5−5x3+4x

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

Determine the value that $f ( x ) = \frac { 3 x - 5 } { x - 6 }$ approaches as $x$ increases and decreases in magnitude without bound.

Find the zeros of the function below algebraically, if any exist. $f ( t ) = \frac { 1 } { 6 } t ^ { 4 } - \frac { 27 } { 2 }$

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

Simplify $f$ below and find any vertical asymptotes of $f$ . $f ( x ) = \frac { x ^ { 2 } - 25 } { x + 5 }$

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

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

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

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

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

Find all the rational zeros of the function $f ( x ) = - 2 x ^ { 5 } - 11 x ^ { 4 } - 19 x ^ { 3 } - 17 x ^ { 2 } - 17 x - 6$ .

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

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

Using the factors $( - 5 x + 2 )$ and $( x - 1 )$ , find the remaining factor(s) of $f ( x ) = 10 x ^ { 4 } + 31 x ^ { 3 } - 84 x ^ { 2 } + 53 x - 10$ and write the polynomial in fully factored form.

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

Find the zeros of the function below algebraically, if any exist. $f ( x ) = x ^ { 5 } - 5 x ^ { 3 } + 4 x$