Deck 4: Polynomial and Rational Functions

Question 1

Use the leading-term test to match the function with the correct graph.
$$f ( x ) = - 0.6 x ^ { 6 } - x ^ { 5 } + 5 x ^ { 4 } - 3 x ^ { 3 } - 6 x ^ { 2 } + x - 3$$

A)

B)

C)

D)

Answer 1

D

Question 2

Use the leading-term test to match the function with the correct graph.
$$f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 15 x ^ { 2 } + x - 14$$

A)

B)

C)

D)

Answer 2

D

Question 3

Find the zeros of the polynomial function and state the multiplicity of each.
$$f ( x ) = x ^ { 3 } + 8 x ^ { 2 } - x - 8$$

A) $- 1$ , multiplicity $1 ; 1$ , multiplicity $1 ; 8$ , multiplicity 1
B) $- 8$ , multiplicity $2 ; 1$ , multiplicity 1
C) $- 8$ , multiplicity $1 ; - 1$ , multiplicity $1 ; 1$ , multiplicity 1
D) $- 8$ , multiplicity $1 ; - 1$ , multiplicity 2

Answer 3

C

Question 4

Solve.
$$\mathrm { x } ^ { 2 } > 25$$

A) $( - \infty , - 5 ) \cup ( 5 , \infty )$
B) $( - 5 , \infty )$
C) $( - 5,5 )$
D) $( 5 , \infty )$

Question 5

Use the leading-term test to match the function with the correct graph.
$$f ( x ) = 2 x ^ { 2 } + 6$$

A)

B)

C)

D)

Question 6

Match the equation with the appropriate graph.
$$f ( x ) = \frac { 18 } { x ^ { 2 } + 9 }$$

A)

B)

C)

D)

Question 7

Match the equation with the appropriate graph.
$$f ( x ) = \frac { 2 x ^ { 2 } } { x ^ { 2 } - 4 }$$

A)

B)

C)

D)

Question 8

Match the equation with the appropriate graph.
$$f ( x ) = \frac { x ^ { 3 } } { x ^ { 2 } + 9 }$$

A)

B)

C)

D)

Question 9

Use the leading-term test to match the function with the correct graph.
$$f ( x ) = 4 x + 4 x ^ { 2 } - 9 x ^ { 3 }$$

A)

B)

C)

D)

Question 10

Use the leading-term test to match the function with the correct graph.
$$f ( x ) = - 0.3 x ^ { 7 } + 0.14 x ^ { 6 } - 0.13 x ^ { 5 } + x ^ { 4 } + x ^ { 3 } - 8 x - 5$$

A)

B)

C)

D)

Question 11

Graph the function.
$$f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$$

A)

B)


C)

D)

Question 12

Use the leading-term test to match the function with the correct graph.
$$f ( x ) = x ^ { 5 } - x ^ { 3 } + x ^ { 2 } + 4$$

A)

B)

C)

D)

Question 13

Solve the given inequality (a related function is graphed).
$$\frac { 5 x } { x ^ { 2 } - 9 } \geq 0$$

A) $( - 3,0 ) \cup ( 3 , \infty )$
B) $( - 3,0 ] \cup ( 3 , \infty )$
C) $( - 3 , \infty )$
D) $( - \infty , - 3 ) \cup [ 0,3 )$

Question 14

Use the leading-term test to match the function with the correct graph.
$$f ( x ) = - \frac { 1 } { 3 } x ^ { 3 } - 6 x ^ { 2 } + 8 x + 33$$

A)

B)

C)

D)

Question 15

Find the correct end behavior diagram for the given polynomial function.
$$f ( x ) = - x ^ { 5 } - 5 x ^ { 3 } - 2 x + 1$$

Question 16

Use the leading-term test to match the function with the correct graph.
$$f ( x ) = \frac { 1 } { 4 } x ^ { 2 } - 5$$

A)

B)

C)

D)

Question 17

Match the equation with the appropriate graph.
$$f ( x ) = \frac { 18 } { x ^ { 2 } - 9 }$$

A)

B)

C)

D)

Question 18

Match the equation with the appropriate graph.
$$f ( x ) = \frac { 6 x } { x ^ { 2 } - 1 }$$

A)

B)

C)

D)

Question 19

Match the equation with the appropriate graph.
$$f ( x ) = \frac { x ^ { 3 } } { x ^ { 2 } - 4 }$$

A)

B)

C)

D)

Question 20

Graph the function, showing all asymptotes (those that do not correspond to an axis) as dashed lines. List the x- and
y-intercepts.
$$f ( x ) = \frac { x - 4 } { x + 5 }$$

A) x-intercept: $( 4,0 ) ; y$ -intercept: $\left( 0 , - \frac { 4 } { 5 } \right)$ ;

B) x-intercept: $( 4,0 ) ; y$ -intercept: $\left( 0 , \frac { 4 } { 5 } \right)$

C) $x$ -intercept: $( 4,0 ) ; y$ -intercept: $\left( 0 , - \frac { 4 } { 5 } \right)$ ;

D) $x$ -intercept: $( 4,0 ) ; y$ -intercept: $\left( 0 , \frac { 4 } { 5 } \right)$ ;

Question 21

Solve.
The population $\mathrm { P }$ , in thousands, of Pine Grove is given by $\mathrm { P } ( \mathrm { t } ) = \frac { 600 \mathrm { t } } { 2 \mathrm { t } ^ { 2 } + 9 }$ , where $t$ is the time, in months. Find the interval on which the population was 40 thousand or greater.

A) $[ 0.526,5.474 ]$
B) $( - \infty , \infty )$
C) $[ 0.658,6.842 ]$
D) $[ 5.474 , \infty )$

Question 22

Solve the problem.
Assume that a person's threshold weight $W$ , defined as the weight above which the risk of death rises dramatically, is given by $W ( h ) = \left( \frac { h } { 12.3 } \right) ^ { 3 }$ , where $W$ is in pounds and $h$ is the person's height in inches. Find the threshold weight for a person who is $6 \mathrm { ft } 1$ in. tall. Round your answer to the nearest pound.

A) $122 \mathrm { lb }$
B) $209.1 \mathrm { lb }$
C) $221.5 \mathrm { lb }$
D) $235.9 \mathrm { lb }$

Question 23

Solve the problem.
The position of an object moving in a straight line is given by $s = 6 t ^ { 2 } - 4 t$ , where $s$ is in meters and $t$ is the time in seconds the object has been in motion. How far will an object move in 20 seconds?

A) $40 \mathrm {~m}$
B) $800 \mathrm {~m}$
C) $1480 \mathrm {~m}$
D) $2320 \mathrm {~m}$

Question 24

Find the oblique asymptote, if any, of the rational function.
$$f ( x ) = \frac { x ^ { 3 } - 5 x ^ { 2 } + 4 x - 1 } { x ^ { 2 } + 4 x }$$

A) $y = x - 9$
B) $y = x + 9$
C) None
D) $y = x$

Question 25

Use synthetic division to find the function value.
$$f ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 5 x - 4 ; \text { find } f ( 5 )$$

A) $- 150$
B) 104
C) $- 154$
D) 96

Question 26

Solve.
$$x ^ { 3 } - 2 x ^ { 2 } < 29 x - 30$$

A) $( - \infty , 1 ) \cup ( 6 , \infty )$
B) $( - \infty , 1 )$
C) $( - 5,1 ) \cup ( 6 , \infty )$
D) $( - \infty , - 5 ) \cup ( 1,6 )$

Question 27

Using synthetic division, determine whether the numbers are zeros of the polynomial.
$$- 3,0 ; f ( x ) = - 4 x ^ { 3 } + x ^ { 2 } + 2 x + 7$$

A) No; no
B) No; yes
C) Yes; no
D) Yes; yes

Question 28

Solve the problem.
If there are $x$ teams in a sports league and all the teams play each other twice, a total of $N ( x )$ games are played, where $N ( x ) = x ^ { 2 } - x$ . A soccer league has 6 teams, each of which plays the others twice. If the league pays $\$ 49$ per game for the field and officials, how much will it cost to play the entire schedule?

A) $\$ 2058$
B) $\$ 1470$
C) $\$ 1764$
D) $\$ 1568$

Question 29

Find the zeros of the polynomial function and state the multiplicity of each.
$$f ( x ) = x ^ { 3 } + x ^ { 2 } - 3 x - 3$$

A) $- 1$ , multiplicity $1 ; 1$ , multiplicity $1 ; \sqrt { 3 }$ , multiplicity 1
B) $- 1$ , multiplicity 1 ; 1 , multiplicity 1 ; 5 , multiplicity 1
C) $- 1$ , multiplicity $1 ; 1$ , multiplicity $1 ; \sqrt { 3 }$ , multiplicity $1 ; - \sqrt { 3 }$ , multiplicity 1
D) $- 1$ , multiplicity $1 ; \sqrt { 3 }$ , multiplicity $1 ; - \sqrt { 3 }$ , multiplicity 1

Question 30

Graph the function, showing all asymptotes (those that do not correspond to an axis) as dashed lines. List the x- and
y-intercepts.
$$f ( x ) = \frac { 1 } { x - 2 }$$

$\text { A) No x-intercepts, } y \text {-intercept: }\left(0, \frac{1}{2}\right) \text {; }$

$\text { B) No x-intercepts, } y \text {-intercept: }\left(0, \frac{1}{2}\right) \text {; }$


$$\text { C) No } x \text {-intercepts, } y \text {-intercept: } \left( 0 , - \frac { 1 } { 2 } \right) \text {; }$$

D) $\text { No } x \text {-intercepts, } y \text {-intercept: }\left(0, \frac{1}{3}\right)$

Question 31

Find the requested polynomial.
Find a polynomial function of degree 3 with 2, i, -i as zeros. A) $f ( x ) = x ^ { 3 } - 2 x ^ { 2 } + x - 2$
B) $f ( x ) = x ^ { 3 } - 2 i x ^ { 2 } + x - 2 i$
C) $f ( x ) = x ^ { 3 } - 2 x ^ { 2 } + i x - 2 i$
D) $f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - x + 2$

Question 32

Find the correct end behavior diagram for the given polynomial function.
$$f ( x ) = - x ^ { 6 } + 3 x ^ { 5 } - x ^ { 2 } - 4 x + 3$$

Question 33

Solve.
An open-top rectangular box has a square base and it will hold 103 cubic centimeters (cc). Each side of the base has length $x \mathrm {~cm}$ , and the box has a height of $y \mathrm {~cm}$ . Express the surface area $S$ as a function of the length $x$ of a side of the base.

A) $S ( x ) = \frac { 412 } { x } + x ^ { 2 }$
B) $S ( x ) = \frac { 206 } { x } + x ^ { 2 }$
C) $S ( x ) = 412 + x ^ { 2 }$
D) $S ( x ) = \frac { 103 } { x } + x ^ { 2 }$

Question 34

Sketch the graph of the polynomial function. Use the rational zeros theorem when finding the zeros.
$$f ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 13 x + 6$$

A)

B)

C)

D)

Question 35

Solve.
The profit made when $t$ units are sold is given by $P = t ^ { 2 } - 24 t + 140$ for $t > 0$ . Determine the values of $t$ for which $P < 0$ (a loss is taken).

A) $( 10,14 )$
B) $( 0 , \infty )$
C) $[ 10,14 ]$
D) $( - \infty , 10 ) \cup ( 14 , \infty )$

Question 36

Use long division to determine whether the binomial is a factor of f(x).
$$f ( x ) = x ^ { 4 } - x ^ { 3 } - 3 x ^ { 2 } + 4 x + 7 ; x + 2$$

A) Yes
B) $\mathrm { No }$

Question 37

List the critical values of the related function. Then solve the inequality.
$$\frac { x } { x ^ { 2 } + 3 x - 4 } + \frac { 2 } { x ^ { 2 } - 16 } \leq \frac { 2 x } { x ^ { 2 } - 5 x + 4 }$$

A) $- 5 - \sqrt { 23 } , - 4 , - 5 + \sqrt { 23 } , 1,4 ; [ - 5 - \sqrt { 23 } , - 4 ) \cup [ - 5 + \sqrt { 23 } , 1 ) \cup ( 4 , \infty )$
B) $- 4 - \sqrt { 23 } , - 5 , - 4 + \sqrt { 23 } , 1,5 ; [ - 4 - \sqrt { 23 } , - 5 ) \cup [ - 4 + \sqrt { 23 } , 1 ) \cup ( 5 , \infty )$
C) No critical values; $\varnothing$
D) $- 5 - \sqrt { 33 } , - 4 , - 5 + \sqrt { 33 } , 1,4 ; [ - 5 - \sqrt { 33 } , - 4 ) \cup [ - 5 + \sqrt { 33 } , 1 ) \cup ( 4 , \infty )$

Question 38

Solve the inequality.
For the function $h ( x ) = \frac { 2 x } { ( x - 2 ) ( x - 6 ) }$ , solve $h ( x ) < 0$ .

A) $( - \infty , 2 ) \cup ( 6 , \infty )$
B) $( - \infty , 0 ) \cup ( 2,6 )$
C) $[ 2,6 ]$
D) $( - \infty , 2 ] \cup [ 6 , \infty )$

Question 39

Solve.
$x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20 \geq 0$

A) $[ - 5 , \infty )$
B) $[ - 5 , - 2 ] \cup [ 2 , \infty )$
C) $[ - 2,2 ] \cup [ 5 , \infty )$
D) $[ - 5,2 ]$

Question 40

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative
real zeros for the function.
$$P ( x ) = - 5 x ^ { 4 } + 4 x ^ { 3 } - 7 x ^ { 2 } + 5 x - 4$$

A) 0 , 2, or 4 positive; 0 , 2 , or 4 negative
B) 0 or 2 positive; 0,2 , or 4 negative
C) 0 or 2 positive; 0 negative
D) 0,2 , or 4 positive; 0 negative

Question 41

Solve.
$$x ^ { 3 } + 10 x ^ { 2 } + 23 x \geq 0$$

A) $[ - 5 - \sqrt { 2 } , - 5 + \sqrt { 2 } ] \cup [ 0 , \infty )$
B) $( - \infty , - 5 - \sqrt { 2 } ] \cup [ 0 , \infty ]$
C) $[ - 5 - \sqrt { 2 } , \infty )$
D) $( - \infty , - 5 - \sqrt { 2 } ] \cup [ - 5 + \sqrt { 2 } , \infty )$

Question 42

Use substitution to determine whether the given number is a zero of the given polynomial.
3; $f ( x ) = - x ^ { 4 } - 8 x ^ { 2 } - 5 x + 168$

A) Yes
B) $\mathrm { No }$

Question 43

Find the horizontal asymptote, if any, of the rational function.
$$f ( x ) = \frac { x ^ { 2 } + 5 x - 7 } { x - 7 }$$

A) $y = 1$
B) $y = - 5$
C) $y = 0$
D) None

Question 44

Classify the polynomial as constant, linear, quadratic, cubic, or quartic, and determine the leading term, the leading
coefficient, and the degree of the polynomial.
$$f ( x ) = - 13 - x ^ { 2 }$$

A) Constant; $- 13 ; - 13 ; 0$
B) Linear; $- x ^ { 2 } ; - 1 ; 1$
C) Linear; $- 13 ; - 13 ; 1$
D) Quadratic; $- x ^ { 2 } ; - 1 ; 2$

Question 45

Use synthetic division to find the function value.
$f ( x ) = x ^ { 3 } + 11$ ; find $f ( 2 + i )$ .

A) $2 + 11 i$
B) $2 + 12 \mathrm { i }$
C) $13 + 12 \mathrm { i }$
D) $13 + 11 \mathrm { i }$

Question 46

Solve.
There are $n$ people in a room. The number $N$ of possible handshakes by all the people in the room is given by the function $N ( n ) = \frac { n ( n - 1 ) } { 2 }$ . For what number n of people is $91 \leq N \leq 190$ ?

A) $\{ n \mid 13 \leq n \leq 22 \}$
B) $\{ n \mid 15 \leq n \leq 19 \}$
C) $\{ \mathrm { n } \mid 16 \leq \mathrm { n } \leq 19 \}$
D) $\{ n \mid 14 \leq n \leq 20 \}$

Question 47

Graph the polynomial function. Use synthetic division and the remainder theorem to find the zeros.
$$f ( x ) = - x ^ { 3 } + 2 x ^ { 2 } + 39 x + 72$$

A) $- 3,3,8$ ;

B) $- 3,3 , - 8$ ;

C) $- 3$ (multiplicity 2), 8 ;

D) $- 3$ (multiplicity 2), $- 8$ ;

Question 48

Find the horizontal asymptote, if any, of the rational function.
$$f ( x ) = \frac { 3 x ^ { 3 } - 9 x - 7 } { 7 x ^ { 3 } - 5 x + 2 }$$

A) $y = \frac { 3 } { 7 }$
B) $y = \frac { 9 } { 5 }$
C) $\mathrm { y } = 0$
D) None

Question 49

Given that the polynomial function has the given zero, find the other zeros.
$$f ( x ) = x ^ { 3 } - 2 x ^ { 2 } - 11 x + 52 ; - 4$$

A) $3 + 4 \mathrm { i } , 3 - 4 \mathrm { i }$
B) $3 + 2 i , 3 - 2 i$
C) $1 + 2 \sqrt { 13 } i , 1 - 2 \sqrt { 13 } i$
D) $1 + 2 i , 1 - 2 i$

Question 50

Solve.
$$x ^ { 2 } - 4 x - 12 < 0$$

A) $( - 2,6 )$
B) $( 6 , \infty )$
C) $( - \infty , - 2 ) \cup ( 6 , \infty )$
D) $( - \infty , - 2 )$

Question 51

Use synthetic division to find the quotient and the remainder.
$$\left( 3 x ^ { 4 } + 2 x ^ { 2 } - 1 \right) \div \left( x + \frac { 1 } { 2 } \right)$$

A) $Q ( x ) = 3 x ^ { 3 } + \frac { 3 } { 2 } x ^ { 2 } - \frac { 11 } { 8 } x + \frac { 11 } { 8 } ; R ( x ) = - \frac { 5 } { 16 }$
B) $Q ( x ) = 3 x ^ { 3 } - 1 ; R ( x ) = - 2$
C) $Q ( x ) = 3 x ^ { 3 } - 1 ; R ( x ) = 0$
D) $Q ( x ) = 3 x ^ { 3 } - \frac { 3 } { 2 } x ^ { 2 } + \frac { 11 } { 4 } x - \frac { 11 } { 8 } ; R ( x ) = - \frac { 5 } { 16 }$

Question 52

Solve.
$$x ^ { 2 } + 6 < 2 x$$

A) $\{ 2 \}$
B) $\varnothing$
C) $( - \infty , - 2 )$
D) $( 2 , \infty )$

Question 53

Find only the rational zeros.
$$f ( x ) = x ^ { 4 } - 8 x ^ { 3 } + 4 x ^ { 2 } + 24 x - 21$$

A) $- 7,1$
B) 7,1
C) $7 , - 1$
D) No rational zeros

Question 54

Graph the function, showing all asymptotes (those that do not correspond to an axis) as dashed lines. List the x- and
y-intercepts.
$$f ( x ) = \frac { x ^ { 2 } - 9 } { x + 2 }$$

A) $x$ -intercepts: $( - 3,0 )$ and $( 3,0 ) , y$ -intercept: $\left( 0 , \frac { 9 } { 2 } \right)$

B) $x$ -intercepts: $( - 3,0 )$ and $( 3,0 ) , \mathrm { y }$ -intercept: $\left( 0 , - \frac { 9 } { 2 } \right)$

C) x-intercept: $( 0,0 ) , y$ -intercept: $( 0,0 )$ ;

D) $x$ -intercept: $( 0,0 ) , y$ -intercept: $( 0,0 )$ ;

Question 55

Find the zeros of the polynomial function and state the multiplicity of each.
$$f ( x ) = ( x + 2 ) ^ { 2 } ( x - 1 )$$

A) 2 , multiplicity $2 ; 1$ , multiplicity 1
B) $- 2$ , multiplicity $1 ; 1$ , multiplicity 1
C) -2, multiplicity $1 ; 1$ , multiplicity 2
D) $- 2$ , multiplicity $2 ; 1$ , multiplicity 1

Question 56

Find the correct end behavior diagram for the given polynomial function.
$$f ( x ) = \sqrt { 3 } x ^ { 6 } - x ^ { 5 } + 7 x ^ { 2 } - 4$$

A)
B)
C)
D)

Question 57

For the function find the maximum number of real zeros that the function can have, the maximum number of x-intercepts
that the function can have, and the maximum number of turning points that the graph of the function can have.
$$f ( x ) = 8 x ^ { 3 } + 8 x ^ { 2 } - 8 x + 7$$

A) $3 ; 3 ; 2$
B) $3 ; 2 ; 1$
C) $3 ; 3 ; 3$
D) $2 ; 2 ; 1$

Question 58

Classify the polynomial as constant, linear, quadratic, cubic, or quartic, and determine the leading term, the leading
coefficient, and the degree of the polynomial.
$$f ( x ) = 7 x ^ { 2 } - 10 + 0.12 x - 7 x ^ { 3 }$$

A) Constant; $- 10 ; - 10 ; 1$
B) Cubic; $- 7 x ^ { 3 } ; - 7 ; 3$
C) Cubic; $x ^ { 3 } ; - 7 ; 3$
D) Quadratic; $7 x ^ { 2 } ; 7 ; 2$

Question 59

Use substitution to determine whether the given number is a zero of the given polynomial.
$$- 3 ; f ( x ) = x ^ { 4 } - 3 x ^ { 2 } - 54$$

A) Yes
B) No

Question 60

Solve.
$$x ^ { 2 } + 10 x + 25 \leq 0$$

A) $[ 5 , \infty )$
B) $\{ 5 \}$
C) $( - \infty , - 5 ] \cup [ - 5 , \infty )$
D) $\{ - 5 \}$

Question 61

State the domain of the rational function.
$$f ( x ) = \frac { x ^ { 2 } + 4 x - 12 } { x ^ { 2 } - 2 x - 8 }$$

A) $( - \infty , - 4 ) \cup ( - 4,2 ) \cup ( 2 , \infty )$
B) $( - \infty , - 2 ) \cup ( - 2,4 ) \cup ( 4 , \infty )$
C) $( - \infty , \infty )$
D) $( - \infty , 4 ) \cup ( 4 , \infty )$

Question 62

Solve.
An open-top rectangular box has a square base and it will hold 256 cubic centimeters (cc). Each side of the base has length $x \mathrm {~cm}$ . The box's surface area $S$ is given by
$$S ( x ) = \frac { 1024 } { x } + x ^ { 2 } .$$
Estimate the minimum surface area and the value of $x$ that will yield it.

A) $256 \mathrm {~cm} ^ { 2 }$ when $x = 6 \mathrm {~cm}$
B) $207 \mathrm {~cm} ^ { 2 }$ when $x = 6 \mathrm {~cm}$
C) $256 \mathrm {~cm} ^ { 2 }$ when $x = 8 \mathrm {~cm}$
D) $192 \mathrm {~cm} ^ { 2 }$ when $\mathrm { x } = 8 \mathrm {~cm}$

Question 63

Find only the rational zeros.
$$f ( x ) = x ^ { 5 } - 5 x ^ { 4 } + 5 x ^ { 3 } + 15 x ^ { 2 } - 36 x + 20$$

A) $- 1,2 , - 2$
B) $1,2,3$
C) $1,2 , - 2$
D) No rational zeros

Question 64

Find the horizontal asymptote, if any, of the rational function.
$$f ( x ) = \frac { 5 } { x ^ { 2 } + 2 }$$

A) $y = 5$
B) $y = 0$
C) $x = 0$
D) None

Question 65

Provide the requested response.
Suppose that a polynomial function of degree 5 with rational coefficients has $- 5,4 , - 3,5 - i$ as zeros. Find the other zero(s).

A) $5 + i$
B) $- 5 - \mathrm { i }$
C) $5 , - 4,3,5 + \mathrm { i }$
D) $- 5 + \mathrm { i }$

Question 66

Use long division to determine whether the binomial is a factor of f(x).
$$f ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 31 x + 70 ; x + 5$$

A) Yes
B) $\mathrm { No }$

Question 67

For the function find the maximum number of real zeros that the function can have, the maximum number of x-intercepts
that the function can have, and the maximum number of turning points that the graph of the function can have.
$$f ( x ) = - x ^ { 6 } + 8 x ^ { 5 } - x ^ { 2 } - 4 x + 9$$

A) $6 ; 6 ; 5$
B) $6 ; 5 ; 5$
C) $6 ; 6 ; 6$
D) $5 ; 5 ; 6$

Question 68

List the critical values of the related function. Then solve the inequality.
$$\frac { 2 } { x ^ { 2 } + 1 } \geq \frac { 4 } { 4 x ^ { 2 } + 3 }$$

A) No critical values; $( - \infty , \infty )$
B) $2 ; ( - \infty , 2 ]$
C) No critical values; $\varnothing$
D) $4 ; [ 4 , \infty )$

Question 69

Factor the polynomial f(x). Then solve the equation f(x) = 0.
$$f ( x ) = x ^ { 3 } + 11 x ^ { 2 } + 36 x + 36$$

A) $( x + 2 ) ( x + 3 ) ( x + 6 ) ; 2,3,6$
B) $( x + 2 ) ( x + 3 ) ( x + 6 ) ; - 2 , - 3 , - 6$
C) $( x - 3 ) ( x - 4 ) ( x - 7 ) ; 3,4,7$
D) $( x + 3 ) ( x + 4 ) ( x + 7 ) ; 3,4,7$

Question 70

Graph the function, showing all asymptotes (those that do not correspond to an axis) as dashed lines. List the x- and
y-intercepts.
$$f ( x ) = \frac { x ^ { 2 } - 3 x - 4 } { 3 x ^ { 2 } + 2 }$$

A) No $x$ -intercepts, $y$ -intercept: $( 0 , - 4 )$ ;

B) No $x$ -intercepts, $y$ -intercept: $( 0 , - 4 )$ ;

C) x-intercepts: $( 1,0 )$ and $( - 4,0 )$ ,

D) x-intercepts: $( - 1,0 )$ and $( 4,0 )$ , y-intercept: $( 0 , - 2 )$ ; $\mathrm { y }$ -intercept: $( 0 , - 2 )$ ;

Question 71

List the critical values of the related function. Then solve the inequality.
$$\frac { 1 } { x - 4 } \leq 0$$

A) $- 4,4 ; [ - 4,4 ]$
B) $4 ; ( - \infty , 4 )$
C) No critical values; $\varnothing$
D) $- 4 ; ( - \infty , - 4 ]$

Question 72

Graph the function.
$$h ( x ) = x ^ { 5 } - 17 x ^ { 2 } + 16 x$$

A)

B)


C)

D)

Question 73

Provide the requested response.
Suppose that a polynomial function of degree 4 with rational coefficients has 6, 4, 3i as zeros. Find the other zero.

A)3- i
B)- 3i
C)-6 $$\text { D) } \sqrt { 3 } \mathrm { i }$$

Question 74

Solve.
The average cost per tape, in dollars, for a company to produce x sports videotapes is given by the function $$A ( x ) = \frac { 14 x + 80 } { x } \text { for } x > 0$$
Graph the function on the interval $( 0 , \infty )$ and complete the following:
$$\mathrm { A } ( \mathrm { x } ) \rightarrowــــــــــ \quad \text { as } \mathrm { x } \rightarrow \infty \text {. }$$ A)

$\mathrm { A } ( \mathrm { x } ) \rightarrow 1$ as $\mathrm { x } \rightarrow \infty$ .
B)


$\mathrm { A } ( \mathrm { x } ) \rightarrow 0$ as $\mathrm { x } \rightarrow \infty .$
C)

$\mathrm { A } ( \mathrm { x } ) \rightarrow 19$ as $\mathrm { x } \rightarrow \infty .$
D)

$\mathrm { A } ( \mathrm { x } ) \rightarrow 14$ as $\mathrm { x } \rightarrow \infty$ .

Question 75

Find a rational function that satisfies the given conditions. Answers may vary, but try to give the simplest answer
possible.
Oblique asymptote $y = x + 5$

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

Question 76

Classify the polynomial as constant, linear, quadratic, cubic, or quartic, and determine the leading term, the leading
coefficient, and the degree of the polynomial.
$$g ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 12 x + 8$$

A) Linear; $- 12 x ; - 12 ; 1$
B) Quadratic; $\frac { 1 } { 2 } x ^ { 2 } ; \frac { 1 } { 2 } ; 2$
C) Cubic; $\frac { 1 } { 2 } x ^ { 3 } ; \frac { 1 } { 2 } ; 3$
D) Linear; $8 ; 8 ; 1$

Question 77

Given that the polynomial function has the given zero, find the other zeros.
$$f ( x ) = x ^ { 4 } - 21 x ^ { 2 } - 100 ; - 2 i$$

A) $2 \mathrm { i } , 5 \mathrm { i } , - 5 \mathrm { i }$
B) $2 \mathrm { i } , 10 , - 10$
C) $2 \mathrm { i } , 5 , - 5$
D) $2 \mathrm { i } , 10 \mathrm { i } , - 10 \mathrm { i }$

Question 78

Given that the polynomial function has the given zero, find the other zeros.
$$f ( x ) = x ^ { 3 } - 64 ; 4$$

A) $- 2 + \sqrt { 3 } i , - 2 - \sqrt { 3 } i$
B) $- 2 + 2 \sqrt { 3 } i , - 2 - 2 \sqrt { 3 } i$
C) $- 1 + 2 \sqrt { 3 } i , - 1 - 2 \sqrt { 3 } i$
D) $- 2 + 4 \sqrt { 3 } i , - 2 - 4 \sqrt { 3 } i$

Question 79

Solve the inequality.
For the function $f ( x ) = x ^ { 2 } - 2 x - 35$ , solve $f ( x ) \leq 0$ .

A) $( - \infty , - 5 ] \cup [ 7 , \infty )$
B) $[ - 5,7 ]$
C) $( - \infty , - 5 ]$
D) $[ 7 , \infty )$

Question 80

Use synthetic division to find the function value.
$f ( x ) = x ^ { 5 } - 10 x ^ { 4 } + 15 x ^ { 3 } - 4 x - 300 ;$ find $f ( 2 )$

A) $- 314$
B) 284
C) $- 318$
D) $- 316$