Question 1

Determine whether the statement is true or false.
-$$\text { The function } f ( x ) = \frac { x + 1 } { x ^ { 2 } } \text { is an even function. }$$

Accepted Answer

A) True 
 B)False

Question 2

Solve the problem
-John owns a hot dog stand. He has found that his profit is represented by the equation $\mathrm { P } ( \mathrm { x } ) = - \mathrm { x } ^ { 2 } + 70 \mathrm { x } + 82$, with $\mathrm { P }$ being profits and $\mathrm { x }$ the number of hot dogs sold. How many hot dogs must he sell to earn the most profit?

Accepted Answer

A)  35 hot dogs 
B)  47 hot dogs 
C)  23 hot dogs 
D)  36 hot dogs 
A)  35 hot dogs 
B)  47 hot dogs 
C)  23 hot dogs 
D)  36 hot dogs

Question 3

Use the remainder theorem and synthetic division to find f(k).
-$$k = 3 ; f ( x ) = x ^ { 2 } - 3 x + 2$$

Accepted Answer

A)  -20 
B)  2 
C)  -16 
D)  -2 
A)  -20 
B)  2 
C)  -16 
D)  -2

Question 4

Round your answer to two decimal places.
-The period of vibration P for a pendulum varies directly as the square root of the length L. If the period of vibration is 3.5 sec when the length is 49 inches, what is the period when L = 5.0625
Inches?

Accepted Answer

A)  1.125 sec 
B)  4 sec 
C)  3.75 sec 
D)  4.5 sec 
A)  1.125 sec 
B)  4 sec 
C)  3.75 sec 
D)  4.5 sec

Question 5

Round to the nearest tenth unless indicated otherwise.
-The volume of wood in a tree varies jointly as the height of the tree and the square of the distance around the tree trunk. If the volume of wood is $15.84$ cubic feet when the height is 22 feet and the distance around the trunk is 3 feet, what is the volume of wood obtained from a tree that is 30 feet tall having a measurement of 5 feet around the trunk?

Accepted Answer

A)  $64.0 \mathrm { ft } ^ { 3 }$ 
B)  $60.0 \mathrm { ft } ^ { 3 }$ 
C)  $52.0 \mathrm { ft } ^ { 3 }$ 
D)  $69.0 \mathrm { ft } ^ { 3 }$ 
A)  $64.0 \mathrm { ft } ^ { 3 }$ 
B)  $60.0 \mathrm { ft } ^ { 3 }$ 
C)  $52.0 \mathrm { ft } ^ { 3 }$ 
D)  $69.0 \mathrm { ft } ^ { 3 }$

Question 6

Translate the given formula to an English phrase using the word "varies".
-If $m$ varies directly as $x$ and $y$, and $m = 24$ when $x = 6$ and $y = 9$, find $m$ when $x = 15$ and $y = 8$.

Accepted Answer

A)  $\frac { 140 } { 3 }$ 
B)  $\frac { 160 } { 3 }$ 
C)  60 
D)  $\frac { 64 } { 3 }$ 
A)  $\frac { 140 } { 3 }$ 
B)  $\frac { 160 } { 3 }$ 
C)  60 
D)  $\frac { 64 } { 3 }$

Question 7

Tell whether a linear model or a quadratic model is appropriate for the data. If linear, tell whether the slope should be
positive or negative. If quadratic, decide whether the leading coefficient a of $x ^ { 2 }$ should be positive or negative.
-The height of a bouncing ball during one bounce, as a function of time.

Accepted Answer

A)  quadratic; negative 
B)  linear; positive 
C)  linear; negative 
D)  quadratic; positive 
A)  quadratic; negative 
B)  linear; positive 
C)  linear; negative 
D)  quadratic; positive

Question 8

Determine which of the rational functions given below has the following feature(s).
-The horizontal asymptote is $\mathrm { y } = 4$

Accepted Answer

A)  $f ( x ) = \frac { 4 x ^ { 2 } - 1 } { x + 9 }$ 
B)  $f ( x ) = \frac { 4 } { x - 4 }$ 
C)  $f ( x ) = \frac { x + 4 } { x + 9 }$ 
D)  $f ( x ) = \frac { 4 x - 1 } { x + 9 }$ 
A)  $f ( x ) = \frac { 4 x ^ { 2 } - 1 } { x + 9 }$ 
B)  $f ( x ) = \frac { 4 } { x - 4 }$ 
C)  $f ( x ) = \frac { x + 4 } { x + 9 }$ 
D)  $f ( x ) = \frac { 4 x - 1 } { x + 9 }$

Question 9

Translate the given formula to an English phrase using the word "varies".
-If $\mathrm { y }$ varies directly as $\mathrm { x } ^ { 2 }$ and inversely as $\mathrm { m }$, and $\mathrm { y } = 9$ when $\mathrm { x } = 2$ and $\mathrm { m } = 5$, find $\mathrm { y }$ when $\mathrm { x } = 8$ and $\mathrm { m } = 10$.

Accepted Answer

A)  18 
B)  $0.25$ 
C)  72 
D)  $1.13$ 
A)  18 
B)  $0.25$ 
C)  72 
D)  $1.13$

Question 10

Use the remainder theorem and synthetic division to find f(k).
-$\mathrm { k } = 2 + \mathrm { i } ; \mathrm { f } ( \mathrm { x } ) = \mathrm { x } ^ { 3 } + 10$

Accepted Answer

A)  $12 + 11 \mathrm { i }$ 
B)  $2 + 11 \mathrm { i }$ 
C)  $12 + 12 \mathrm { i }$ 
D)  $2 + 12 \mathrm { i }$ 
A)  $12 + 11 \mathrm { i }$ 
B)  $2 + 11 \mathrm { i }$ 
C)  $12 + 12 \mathrm { i }$ 
D)  $2 + 12 \mathrm { i }$

Question 11

Solve the problem
-The following polynomial approximates the rabbit population in a particular area. $$R ( x ) = - 0.121 x ^ { 5 } + 3.013 x ^ { 4 } + 3500$$ where x is the number of years from year 0. Use a graphing calculator to describe the rabbit population from the years 0 to 25.

Accepted Answer

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

Question 12

Provide an appropriate response.
-$$f ( x ) = \frac { 5 } { x - 1 }$$

Accepted Answer

A)  Approaches $\infty$ from the left and the right 
B)  Approaches $\infty$ from the left, approaches $- \infty$ from the right 
C)  Approaches $- \infty$ from the left and the right 
D)  Approaches $- \infty$ from the left, approaches $\infty$ from the right 
A)  Approaches $\infty$ from the left and the right 
B)  Approaches $\infty$ from the left, approaches $- \infty$ from the right 
C)  Approaches $- \infty$ from the left and the right 
D)  Approaches $- \infty$ from the left, approaches $\infty$ from the right

Question 13

Graph the polynomial function. Factor first if the expression is not in factored form.
-$f(x)=(3 x+1)(x+2)^{2}$

Accepted Answer

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

Question 14

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
-Zeros of $3 , i , - i$ and $P ( 2 ) = 25$

Accepted Answer

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

Question 15

Use a graphing calculator to approximate the real zeros. Give each zero as a decimal to the nearest tenth.
-$$f ( x ) = x ^ { 3 } + x ^ { 2 } - 11 x - 11$$

Accepted Answer

A)  $1,5.5 , - 5.5$ 
B)  $- 1,3.3 , - 3.3$ 
C)  $1,3.3 , - 3.3$ 
D)  $1,3.3 \mathrm { i } , - 3.3 \mathrm { i }$ 
A)  $1,5.5 , - 5.5$ 
B)  $- 1,3.3 , - 3.3$ 
C)  $1,3.3 , - 3.3$ 
D)  $1,3.3 \mathrm { i } , - 3.3 \mathrm { i }$

Question 16

Find a polynomial of least degree with only real coefficients and having the given zeros.
-$6 , - 10$, and $2 + 3 \mathrm { i }$

Accepted Answer

A)  $f ( x ) = x ^ { 4 } - 8 x ^ { 3 } - 18 x ^ { 2 } + 146 x - 780$ 
B)  $f ( x ) = x ^ { 4 } - 63 x ^ { 2 } + 292 x - 780$ 
C)  $f ( x ) = x ^ { 4 } - 8 x ^ { 3 } + 18 x ^ { 2 } - 146 x + 780$ 
D)  $f ( x ) = x ^ { 4 } - 73 x ^ { 2 } + 292 x - 780$ 
A)  $f ( x ) = x ^ { 4 } - 8 x ^ { 3 } - 18 x ^ { 2 } + 146 x - 780$ 
B)  $f ( x ) = x ^ { 4 } - 63 x ^ { 2 } + 292 x - 780$ 
C)  $f ( x ) = x ^ { 4 } - 8 x ^ { 3 } + 18 x ^ { 2 } - 146 x + 780$ 
D)  $f ( x ) = x ^ { 4 } - 73 x ^ { 2 } + 292 x - 780$

Question 17

Provide an appropriate response.
-For what values of a does the quadratic function $f ( x ) = a x ^ { 2 } - 6 x + 3$ have no $x$-intercepts?

Accepted Answer

A) $a > 9$ B) $a > 3$ C) $a < 9$ D) $a < 3$ A) $a > 9$ B) $a > 3$ C) $a < 9$ D) $a < 3$

Question 18

Find an equation for the rational function graph.
-

Accepted Answer

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

Question 19

Round to the nearest tenth unless indicated otherwise.
-The volume of a gas varies inversely as the pressure and directly as the temperature (in degrees Kelvin). If a certain gas occupies a volume of $2.1$ liters at a temperature of $340 \mathrm {~K}$ and a pressure of 22 newtons per square centimeter, find the volume when the temperature is $442 \mathrm {~K}$ and the pressure is 11 newtons per square centimeter.

Accepted Answer

A)  $35.5 \mathrm {~L}$ 
B)  $0.8 \mathrm {~L}$ 
C)  $5.5 \mathrm {~L}$ 
D)  $32.5 \mathrm {~L}$ 
A)  $35.5 \mathrm {~L}$ 
B)  $0.8 \mathrm {~L}$ 
C)  $5.5 \mathrm {~L}$ 
D)  $32.5 \mathrm {~L}$

Question 20

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.
-$$f ( x ) = x ^ { 3 } - x ^ { 2 } + 4 ; k = - 2$$

Accepted Answer

A)  $f ( x ) = ( x + 2 ) \left( x ^ { 2 } - 3 x + 6 \right) + 4$ 
B)  $f ( x ) = ( x + 2 ) \left( 3 x ^ { 2 } - 4 x + 2 \right) + 16$ 
C)  $f ( x ) = ( x + 2 ) \left( x ^ { 2 } + x + 2 \right) + 8$ 
D)  $f ( x ) = ( x + 2 ) \left( x ^ { 2 } - 3 x + 6 \right) - 8$ 
A)  $f ( x ) = ( x + 2 ) \left( x ^ { 2 } - 3 x + 6 \right) + 4$ 
B)  $f ( x ) = ( x + 2 ) \left( 3 x ^ { 2 } - 4 x + 2 \right) + 16$ 
C)  $f ( x ) = ( x + 2 ) \left( x ^ { 2 } + x + 2 \right) + 8$ 
D)  $f ( x ) = ( x + 2 ) \left( x ^ { 2 } - 3 x + 6 \right) - 8$

Determine whether the statement is true or false. - $\text { The function } f ( x ) = \frac { x + 1 } { x ^ { 2 } } \text { is an even function. }$

Use the remainder theorem and synthetic division to find f(k). - $k = 3 ; f ( x ) = x ^ { 2 } - 3 x + 2$

Round your answer to two decimal places. -The period of vibration P for a pendulum varies directly as the square root of the length L. If the period of vibration is 3.5 sec when the length is 49 inches, what is the period when L = 5.0625 Inches?

Translate the given formula to an English phrase using the word "varies". -If $m$ varies directly as $x$ and $y$ , and $m = 24$ when $x = 6$ and $y = 9$ , find $m$ when $x = 15$ and $y = 8$ .

Determine which of the rational functions given below has the following feature(s). -The horizontal asymptote is $\mathrm { y } = 4$

Use the remainder theorem and synthetic division to find f(k). - $\mathrm { k } = 2 + \mathrm { i } ; \mathrm { f } ( \mathrm { x } ) = \mathrm { x } ^ { 3 } + 10$

Solve the problem -The following polynomial approximates the rabbit population in a particular area. $R ( x ) = - 0.121 x ^ { 5 } + 3.013 x ^ { 4 } + 3500$ where x is the number of years from year 0. Use a graphing calculator to describe the rabbit population from the years 0 to 25.

Provide an appropriate response. - $f ( x ) = \frac { 5 } { x - 1 }$

Graph the polynomial function. Factor first if the expression is not in factored form. - $f(x)=(3 x+1)(x+2)^{2}$ $Graph the polynomial function. Factor first if the expression is not in factored form. - f(x)=(3 x+1)(x+2)^{2}$

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. -Zeros of $3 , i , - i$ and $P ( 2 ) = 25$

Use a graphing calculator to approximate the real zeros. Give each zero as a decimal to the nearest tenth. - $f ( x ) = x ^ { 3 } + x ^ { 2 } - 11 x - 11$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $6 , - 10$ , and $2 + 3 \mathrm { i }$

Provide an appropriate response. -For what values of a does the quadratic function $f ( x ) = a x ^ { 2 } - 6 x + 3$ have no $x$ -intercepts?

Find an equation for the rational function graph. -

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. - $f ( x ) = x ^ { 3 } - x ^ { 2 } + 4 ; k = - 2$

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

Determine whether the statement is true or false. - The function f(x)=x+1x2 is an even function. \text { The function } f ( x ) = \frac { x + 1 } { x ^ { 2 } } \text { is an even function. } The function f(x)=x2x+1​ is an even function.

Use the remainder theorem and synthetic division to find f(k). - k=3;f(x)=x2−3x+2k = 3 ; f ( x ) = x ^ { 2 } - 3 x + 2k=3;f(x)=x2−3x+2