Question 1

Use the factor theorem to decide whether or not the second polynomial is a factor of the first.
-$8 x ^ { 3 } + 37 x ^ { 2 } - 16 x + 5 ; x + 5$

Accepted Answer

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

Question 2

Determine whether the statement is true or false.
-$$\text { The function } f ( x ) = \frac { 1 } { x } \text { is decreasing on the interval } ( 0 , \infty )$$

Accepted Answer

A) True 
 B)False

Question 3

Solve the problem
-The following polynomial approximates the shark population in a particular area. $$R ( x ) = - 0.018 x ^ { 5 } + 4.013 x ^ { 4 } + 300$$ where x is the number of years from year 0. Use a graphing calculator to describe the shark population from the years 0 to 25.

Accepted Answer

A)  The population remains stable. 
B)  The population decreases. 
C)  The population increases. 
A)  The population remains stable. 
B)  The population decreases. 
C)  The population increases.

Question 4

Determine which of the rational functions given below has the following feature(s).
-x-intercepts: $( 3,0 )$ and $( 4,0 )$, y-intercept: $( 0,12 )$, vertical asymptote: $x = 1$, horizontal asymptote: $y = 1$

Accepted Answer

A)  $f ( x ) = \frac { ( x + 3 ) ( x + 4 ) } { ( x + 1 ) ^ { 2 } }$ 
B)  $f ( x ) = \frac { ( x - 3 ) ( x - 4 ) } { ( x - 1 ) }$ 
C)  $f ( x ) = \frac { ( x + 3 ) ( x + 4 ) } { ( x + 1 ) }$ 
D)  $f ( x ) = \frac { ( x - 3 ) ( x - 4 ) } { ( x - 1 ) ^ { 2 } }$ 
A)  $f ( x ) = \frac { ( x + 3 ) ( x + 4 ) } { ( x + 1 ) ^ { 2 } }$ 
B)  $f ( x ) = \frac { ( x - 3 ) ( x - 4 ) } { ( x - 1 ) }$ 
C)  $f ( x ) = \frac { ( x + 3 ) ( x + 4 ) } { ( x + 1 ) }$ 
D)  $f ( x ) = \frac { ( x - 3 ) ( x - 4 ) } { ( x - 1 ) ^ { 2 } }$

Question 5

Use the remainder theorem and synthetic division to find f(k).
-$$k = - \frac { 1 } { 2 } ; f ( x ) = 6 x ^ { 3 } - 23 x ^ { 2 } - 11 x$$

Accepted Answer

A)  $- 2$ 
B)  2 
C)  0 
D)  $- 1$ 
A)  $- 2$ 
B)  2 
C)  0 
D)  $- 1$

Question 6

Find the equation that the given graph represents.
-

Accepted Answer

A)  $P ( x ) = - x ^ { 3 } + 15 x ^ { 2 } + x - 10$ 
B)  $P ( x ) = x ^ { 5 } + 3 x ^ { 4 } - 5 x ^ { 3 } - 15 x ^ { 2 } + x + 10$ 
C)  $P ( x ) = - x ^ { 5 } - 5 x ^ { 3 } - 6 x ^ { 2 } + 10 x$ 
D)  $P ( x ) = x ^ { 4 } - 5 x ^ { 3 } + 6 x ^ { 2 } + x + 10$ 
A)  $P ( x ) = - x ^ { 3 } + 15 x ^ { 2 } + x - 10$ 
B)  $P ( x ) = x ^ { 5 } + 3 x ^ { 4 } - 5 x ^ { 3 } - 15 x ^ { 2 } + x + 10$ 
C)  $P ( x ) = - x ^ { 5 } - 5 x ^ { 3 } - 6 x ^ { 2 } + 10 x$ 
D)  $P ( x ) = x ^ { 4 } - 5 x ^ { 3 } + 6 x ^ { 2 } + x + 10$

Question 7

Find an equation for the rational function graph.
-

Accepted Answer

A)  $f ( x ) = \frac { 2 ( x + 3 ) ( x + 8 ) } { ( x + 2 ) ( x + 8 ) }$ 
B)  $f ( x ) = \frac { 2 ( x - 3 ) ( x - 8 ) } { ( x - 2 ) ( x - 8 ) }$ 
C)  $f ( x ) = \frac { 2 ( x - 3 ) } { x - 2 }$ 
D)  $f ( x ) = \frac { 2 ( x - 2 ) ( x - 8 ) } { ( x - 3 ) ( x - 8 ) }$ 
A)  $f ( x ) = \frac { 2 ( x + 3 ) ( x + 8 ) } { ( x + 2 ) ( x + 8 ) }$ 
B)  $f ( x ) = \frac { 2 ( x - 3 ) ( x - 8 ) } { ( x - 2 ) ( x - 8 ) }$ 
C)  $f ( x ) = \frac { 2 ( x - 3 ) } { x - 2 }$ 
D)  $f ( x ) = \frac { 2 ( x - 2 ) ( x - 8 ) } { ( x - 3 ) ( x - 8 ) }$

Question 8

Match the equation to the correct graph.
-$$y = \frac { 1 } { 2 } ( x - 5 ) ^ { 2 } + 6$$

Accepted Answer

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

Question 9

Find an equation for the rational function graph.
-In the following formula, $f ( x )$ is the minimum number of hours of studying required to attain a test score of $x : f ( x ) = \frac { 0.44 x } { 100.5 - x }$. How many hours of study are needed to score 94 ?

Accepted Answer

A)  $100.03 \mathrm { hr }$ 
B)  $14.53 \mathrm { hr }$ 
C)  $6.36 \mathrm { hr }$ 
D)  $63.60 \mathrm { hr }$ 
A)  $100.03 \mathrm { hr }$ 
B)  $14.53 \mathrm { hr }$ 
C)  $6.36 \mathrm { hr }$ 
D)  $63.60 \mathrm { hr }$

Question 10

Solve the problem
-$$\begin{array} { c | c } 
& \begin{array} { c } 
\text { Population } \
\text { Year }
\end{array} \
\hline 0 & 65 \
1 & 65.5 \
2 & 67 \
3 & 69 \
4 & 71.5
\end{array}$$ We used these data to develop the quadratic function $$f ( x ) = 0.037214 x ^ { 2 } + 0.364 x + 65$$ , which models the population of the city y in millions in the year x. Use the model to find the estimated population in
Year 10.

Accepted Answer

A)  71,628,000 
B)  1,490,000 
C)  66,427,000 
D)  1,490,000,000,000 
A)  71,628,000 
B)  1,490,000 
C)  66,427,000 
D)  1,490,000,000,000

Question 11

Solve the problem
-A rock is propelled upward from the top of a building 180 feet tall at an initial velocity of 56 feet per second. The function that describes the height of the rocket in terms of time $t$ is $\mathrm { s } ( \mathrm { t } ) = - 16 \mathrm { t } ^ { 2 } + 56 t + 180$. Determine the maximum height that the rock reaches.

Accepted Answer

A)  $212 \mathrm { ft }$ 
B)  $246 \mathrm { ft }$ 
C)  $191 \mathrm { ft }$ 
D)  $229 \mathrm { ft }$ 
A)  $212 \mathrm { ft }$ 
B)  $246 \mathrm { ft }$ 
C)  $191 \mathrm { ft }$ 
D)  $229 \mathrm { ft }$

Question 12

Provide an appropriate response.
-A quadratic equation $f ( x ) = 0$ has a solution $x = 3$. Its graph has vertex $( - 1 , - 16 )$. What is the other solution of the equation?

Accepted Answer

A)  $- 4$ 
B)  $- 5$ 
C)  $- 3$ 
D)  7 
A)  $- 4$ 
B)  $- 5$ 
C)  $- 3$ 
D)  7

Question 13

Solve the problem
-$$\begin{array} { c | c } 
& \text { Population } \
\text { Year } & \text { (in millions of people) } \
\hline 0 & 65 \
1 & 65.5 \
2 & 67 \
3 & 69 \
4 & 71.5
\end{array}$$ We used this data to develop the quadratic function $$f ( x ) = 0.373 x ^ { 2 } + 0.165 x + 65$$ , which models the population of the city y in millions in the year x. Use the model to find the estimated population in
Year 9.

Accepted Answer

A)  1,490,000,000,000 
B)  1,490,000 
C)  65,538,000 
D)  66,822,000 
A)  1,490,000,000,000 
B)  1,490,000 
C)  65,538,000 
D)  66,822,000

Question 14

Match the equation to the correct graph.
-$$y = 2 ( x + 2 ) ^ { 2 } - 2$$

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

Use synthetic division to perform the division.
-$$\frac { x ^ { 5 } - 3 x ^ { 4 } - 21 x ^ { 3 } + 20 x ^ { 2 } - 15 x + 21 } { x - 6 }$$

Accepted Answer

A)  $x ^ { 4 } + 3 x ^ { 3 } - 3 x ^ { 2 } + 2 x - 3 + \frac { 3 } { x - 6 }$ 
B)  $x ^ { 4 } + 3 x ^ { 3 } - 3 x ^ { 2 } + 2 x + 3 + \frac { 5 } { x - 6 }$ 
C)  $x ^ { 3 } + 3 x ^ { 2 } - 3 x + 2 + \frac { 3 } { x - 6 }$ 
D)  $x ^ { 4 } + 3 x ^ { 3 } - 3 x ^ { 2 } + 2 x + 3$ 
A)  $x ^ { 4 } + 3 x ^ { 3 } - 3 x ^ { 2 } + 2 x - 3 + \frac { 3 } { x - 6 }$ 
B)  $x ^ { 4 } + 3 x ^ { 3 } - 3 x ^ { 2 } + 2 x + 3 + \frac { 5 } { x - 6 }$ 
C)  $x ^ { 3 } + 3 x ^ { 2 } - 3 x + 2 + \frac { 3 } { x - 6 }$ 
D)  $x ^ { 4 } + 3 x ^ { 3 } - 3 x ^ { 2 } + 2 x + 3$

Question 17

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k)q(x) + r for
the given value of k.
-$$f ( x ) = 3 x ^ { 3 } - x ^ { 2 } + 2 x + 5 ; k = - 1$$

Accepted Answer

A)  $f ( x ) = ( x + 1 ) \left( 3 x ^ { 2 } + 2 x \right) + 5$ 
B)  $f ( x ) = ( x - 1 ) \left( 3 x ^ { 2 } + 2 x \right) + 5$ 
C)  $f ( x ) = ( x + 1 ) \left( 3 x ^ { 2 } - 4 x + 2 \right) + 7$ 
D)  $f ( x ) = ( x + 1 ) \left( 3 x ^ { 2 } - 4 x + 6 \right) - 1$ 
A)  $f ( x ) = ( x + 1 ) \left( 3 x ^ { 2 } + 2 x \right) + 5$ 
B)  $f ( x ) = ( x - 1 ) \left( 3 x ^ { 2 } + 2 x \right) + 5$ 
C)  $f ( x ) = ( x + 1 ) \left( 3 x ^ { 2 } - 4 x + 2 \right) + 7$ 
D)  $f ( x ) = ( x + 1 ) \left( 3 x ^ { 2 } - 4 x + 6 \right) - 1$

Question 18

Give the equations of any vertical asymptotes.
-$$f ( x ) = \frac { x - 7 } { x ^ { 2 } - 9 }$$

Accepted Answer

A)  $x = - 3$ 
B)  $x = 3 , x = - 3$ 
C)  $x = 3$ 
D)  $x = 7$ 
A)  $x = - 3$ 
B)  $x = 3 , x = - 3$ 
C)  $x = 3$ 
D)  $x = 7$

Question 19

Identify any vertical, horizontal, or oblique asymptotes in the graph of y = f(x). State the domain of f.
-

Accepted Answer

A)  Vertical: $x = 2$; oblique: $y = x - 2 ; ( - \infty , 2 ) \cup ( 2 , \infty )$ 
B)  Vertical: $x = - 2$; oblique: $y = x + 2 ; ( - \infty , 2 ) \cup ( 2 , \infty )$ 
C)  Vertical: $x = - 2$; oblique: $y = x - 2 ; ( - \infty , - 2 ) \cup ( - 2 , \infty )$ 
D)  Vertical: $x = - 2 ;$ oblique: $y = x + 2 ; ( - \infty , - 2 ) \cup ( - 2 , \infty )$ 
A)  Vertical: $x = 2$; oblique: $y = x - 2 ; ( - \infty , 2 ) \cup ( 2 , \infty )$ 
B)  Vertical: $x = - 2$; oblique: $y = x + 2 ; ( - \infty , 2 ) \cup ( 2 , \infty )$ 
C)  Vertical: $x = - 2$; oblique: $y = x - 2 ; ( - \infty , - 2 ) \cup ( - 2 , \infty )$ 
D)  Vertical: $x = - 2 ;$ oblique: $y = x + 2 ; ( - \infty , - 2 ) \cup ( - 2 , \infty )$

Question 20

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition.
-The polynomial $f ( x ) = x ^ { 4 } - 9 x ^ { 3 } - 22 x ^ { 2 }$ has no real zero greater than 8 .

Accepted Answer

A)  Yes, the boundedness theorem shows that the polynomial has no real zero greater than 8 . 
B)  No, the boundedness theorem does not show that the polynomial has no real zero greater than 8 . 
A)  Yes, the boundedness theorem shows that the polynomial has no real zero greater than 8 . 
B)  No, the boundedness theorem does not show that the polynomial has no real zero greater than 8 .

Use the factor theorem to decide whether or not the second polynomial is a factor of the first. - $8 x ^ { 3 } + 37 x ^ { 2 } - 16 x + 5 ; x + 5$

Determine whether the statement is true or false. - $\text { The function } f ( x ) = \frac { 1 } { x } \text { is decreasing on the interval } ( 0 , \infty )$

Solve the problem -The following polynomial approximates the shark population in a particular area. $R ( x ) = - 0.018 x ^ { 5 } + 4.013 x ^ { 4 } + 300$ where x is the number of years from year 0. Use a graphing calculator to describe the shark population from the years 0 to 25.

Determine which of the rational functions given below has the following feature(s). -x-intercepts: $( 3,0 )$ and $( 4,0 )$ , y-intercept: $( 0,12 )$ , vertical asymptote: $x = 1$ , horizontal asymptote: $y = 1$

Use the remainder theorem and synthetic division to find f(k). - $k = - \frac { 1 } { 2 } ; f ( x ) = 6 x ^ { 3 } - 23 x ^ { 2 } - 11 x$

Find the equation that the given graph represents. -

Find an equation for the rational function graph. -

Match the equation to the correct graph. - $y = \frac { 1 } { 2 } ( x - 5 ) ^ { 2 } + 6$

Find an equation for the rational function graph. -In the following formula, $f ( x )$ is the minimum number of hours of studying required to attain a test score of $x : f ( x ) = \frac { 0.44 x } { 100.5 - x }$ . How many hours of study are needed to score 94 ?

Solve the problem - Population Year 0 65 1 65.5 2 67 3 69 4 71.5 We used these data to develop the quadratic function $f ( x ) = 0.037214 x ^ { 2 } + 0.364 x + 65$ , which models the population of the city y in millions in the year x. Use the model to find the estimated population in Year 10.

Provide an appropriate response. -A quadratic equation $f ( x ) = 0$ has a solution $x = 3$ . Its graph has vertex $( - 1 , - 16 )$ . What is the other solution of the equation?

Match the equation to the correct graph. - $y = 2 ( x + 2 ) ^ { 2 } - 2$

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

Use synthetic division to perform the division. - $\frac { x ^ { 5 } - 3 x ^ { 4 } - 21 x ^ { 3 } + 20 x ^ { 2 } - 15 x + 21 } { x - 6 }$

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k. - $f ( x ) = 3 x ^ { 3 } - x ^ { 2 } + 2 x + 5 ; k = - 1$

Give the equations of any vertical asymptotes. - $f ( x ) = \frac { x - 7 } { x ^ { 2 } - 9 }$

Identify any vertical, horizontal, or oblique asymptotes in the graph of y = f(x). State the domain of f. -

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition. -The polynomial $f ( x ) = x ^ { 4 } - 9 x ^ { 3 } - 22 x ^ { 2 }$ has no real zero greater than 8 .

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Filters

Exam 4: Polynomial and Rational Functions

Use the factor theorem to decide whether or not the second polynomial is a factor of the first. - 8x3+37x2−16x+5;x+58 x ^ { 3 } + 37 x ^ { 2 } - 16 x + 5 ; x + 58x3+37x2−16x+5;x+5

Determine whether the statement is true or false. - The function f(x)=1x is decreasing on the interval (0,∞)\text { The function } f ( x ) = \frac { 1 } { x } \text { is decreasing on the interval } ( 0 , \infty ) The function f(x)=x1​ is decreasing on the interval (0,∞)

Determine which of the rational functions given below has the following feature(s). -x-intercepts: (3,0)( 3,0 )(3,0) and (4,0)( 4,0 )(4,0) , y-intercept: (0,12)( 0,12 )(0,12) , vertical asymptote: x=1x = 1x=1 , horizontal asymptote: y=1y = 1y=1

Use the remainder theorem and synthetic division to find f(k). - k=−12;f(x)=6x3−23x2−11xk = - \frac { 1 } { 2 } ; f ( x ) = 6 x ^ { 3 } - 23 x ^ { 2 } - 11 xk=−21​;f(x)=6x3−23x2−11x

Find the equation that the given graph represents. -

Find an equation for the rational function graph. -

Match the equation to the correct graph. - y=12(x−5)2+6y = \frac { 1 } { 2 } ( x - 5 ) ^ { 2 } + 6y=21​(x−5)2+6

Provide an appropriate response. -A quadratic equation f(x)=0f ( x ) = 0f(x)=0 has a solution x=3x = 3x=3 . Its graph has vertex (−1,−16)( - 1 , - 16 )(−1,−16) . What is the other solution of the equation?

Match the equation to the correct graph. - y=2(x+2)2−2y = 2 ( x + 2 ) ^ { 2 } - 2y=2(x+2)2−2

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

Use synthetic division to perform the division. - x5−3x4−21x3+20x2−15x+21x−6\frac { x ^ { 5 } - 3 x ^ { 4 } - 21 x ^ { 3 } + 20 x ^ { 2 } - 15 x + 21 } { x - 6 }x−6x5−3x4−21x3+20x2−15x+21​

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k. - f(x)=3x3−x2+2x+5;k=−1f ( x ) = 3 x ^ { 3 } - x ^ { 2 } + 2 x + 5 ; k = - 1f(x)=3x3−x2+2x+5;k=−1

Give the equations of any vertical asymptotes. - f(x)=x−7x2−9f ( x ) = \frac { x - 7 } { x ^ { 2 } - 9 }f(x)=x2−9x−7​

Identify any vertical, horizontal, or oblique asymptotes in the graph of y = f(x). State the domain of f. -

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition. -The polynomial f(x)=x4−9x3−22x2f ( x ) = x ^ { 4 } - 9 x ^ { 3 } - 22 x ^ { 2 }f(x)=x4−9x3−22x2 has no real zero greater than 8 .

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Filters

Use the factor theorem to decide whether or not the second polynomial is a factor of the first. - $8 x ^ { 3 } + 37 x ^ { 2 } - 16 x + 5 ; x + 5$

Determine whether the statement is true or false. - $\text { The function } f ( x ) = \frac { 1 } { x } \text { is decreasing on the interval } ( 0 , \infty )$

Determine which of the rational functions given below has the following feature(s). -x-intercepts: $( 3,0 )$ and $( 4,0 )$ , y-intercept: $( 0,12 )$ , vertical asymptote: $x = 1$ , horizontal asymptote: $y = 1$

Use the remainder theorem and synthetic division to find f(k). - $k = - \frac { 1 } { 2 } ; f ( x ) = 6 x ^ { 3 } - 23 x ^ { 2 } - 11 x$

Match the equation to the correct graph. - $y = \frac { 1 } { 2 } ( x - 5 ) ^ { 2 } + 6$

Provide an appropriate response. -A quadratic equation $f ( x ) = 0$ has a solution $x = 3$ . Its graph has vertex $( - 1 , - 16 )$ . What is the other solution of the equation?

Match the equation to the correct graph. - $y = 2 ( x + 2 ) ^ { 2 } - 2$

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

Use synthetic division to perform the division. - $\frac { x ^ { 5 } - 3 x ^ { 4 } - 21 x ^ { 3 } + 20 x ^ { 2 } - 15 x + 21 } { x - 6 }$

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k. - $f ( x ) = 3 x ^ { 3 } - x ^ { 2 } + 2 x + 5 ; k = - 1$

Give the equations of any vertical asymptotes. - $f ( x ) = \frac { x - 7 } { x ^ { 2 } - 9 }$

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition. -The polynomial $f ( x ) = x ^ { 4 } - 9 x ^ { 3 } - 22 x ^ { 2 }$ has no real zero greater than 8 .