Deck 4: Polynomial and Rational Functions

Question 1

Match the equation with its graph. $f ( x ) = \frac { 1 } { 20 } \left( x ^ { 5 } - x ^ { 4 } - 5 x ^ { 3 } - x ^ { 2 } - 6 x \right)$

A)
B)
C)
D)
E)

Answer 1

Question 2

Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } + 11 x ^ { 3 } + 30 x ^ { 2 }$ and determine the mutiplicity of each.

A) $x = 0$ , multiplicity 2; $x = - 6$ , multiplicity 1; $x = - 5$ , multiplicity 1
B) $x = 6$ , multiplicity 2; $x = 5$ , multiplicity 2
C) $x = 0$ , multiplicity 2; $x = 6$ , multiplicity 1; $x = 5$ , multiplicity 1
D) $x = - 6$ , multiplicity 2; $x = - 5$ , multiplicity 2
E) $x = 0$ , multiplicity 1; $x = 6$ , multiplicity 1; $x = - 6$ , multiplicity 1; $x = 5$ , multiplicity 1

Answer 2

$x = 0$ , multiplicity 2; $x = - 6$ , multiplicity 1; $x = - 5$ , multiplicity 1

Question 3

The fuel efficiency, in miles per gallon, for a certain midsize car at various speeds, in miles per hour, is given in the table below. $$\begin{array} { | c | c | c | c | } \hline \mathrm { mph } & \mathrm { mpg } & \mathrm { mph } & \mathrm { mpg } \\\hline 25 & 20 & 55 & 31 \\\hline 30 & 24 & 60 & 33 \\\hline 35 & 32 & 65 & 30 \\\hline 40 & 35 & 70 & 24 \\\hline 45 & 36 & 75 & 22 \\\hline 50 & 40 & & \\\hline\end{array}$$ Find a quadratic model for these data.

A) $y = - 0.0344055944 x ^ { 2 } + 3.662377622 x - 55.77622378$
B) $y = 0.0280652681 x ^ { 2 } + 2.897435897 - 38.32867133$
C) $y = - 0.0165034965 x ^ { 2 } + 1.366713287 x + 5.685314685$
D) $y = - 0.0225174825 x ^ { 2 } + 1.915384615 x - 6.664335664$
E) $y = - 0.0258741259 x ^ { 2 } + 2.578321678 x - 28.03496503$

Answer 3

$y = - 0.0258741259 x ^ { 2 } + 2.578321678 x - 28.03496503$

Question 4

The height in feet of a projectile with an initial velocity of 128 feet per second and an initial height of 50 feet is a function of time t, in seconds, given by $h ( t ) = - 16 t ^ { 2 } + 128 t + 50.$ Find the maximum height of the projectile.

A)409 ft
B)306 ft
C)404 ft
D)162 ft
E)498 ft

Question 5

Examine the leading term and determine the far-left and far-right behavior of the graph of the polynomial function $P ( x ) = 5 x ^ { 5 } + 5 x ^ { 3 } - 2 x ^ { 2 } + 8$ .

A)up to the far left, up to the far right
B)up to the far left, down to the far right
C)down to the far left, up to the far right
D)down to the far left, down to the far right
E)cannot be determined

Question 6

Graph the given function. $f ( x ) = ( x - 2 ) ^ { 2 } - 1$

A)
B)
C)
D)
E)

Question 7

Describe the right-hand and the left-hand behavior of the graph of $p ( x ) = 6 x ^ { 5 } + 10 x ^ { 3 } - 27$ .

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 8

Describe the right-hand and the left-hand behavior of the graph of $q ( x ) = - 3 x ^ { 4 } + 7 x ^ { 3 } + 27$ .

A)Because the degree is even and the leading coefficient is negative, the graph falls to the left and falls to the right.
B)Because the degree is even and the leading coefficient is negative, the graph rises to the left and falls to the right.
C)Because the degree is even and the leading coefficient is negative, the graph falls to the left and rises to the right.
D)Because the degree is even and the leading coefficient is negative, the graph rises to the left and rises to the right.
E)Because the degree is odd and the leading coefficient is negative, the graph rises to the left and rises to the right.

Question 9

A farmer has 192 feet of fencing and wants to build two identical pens for his prize-winning pigs. The pens will be arranged as shown. Determine the dimensions of a pen that will maximize its area.

A) $24 ^ { \prime } \times 32 ^ { \prime }$
B) $24 ^ { \prime } \times 64 ^ { \prime }$
C) $17 ^ { \prime } \times 44 ^ { \prime }$
D) $4 ^ { \prime } \times 152 ^ { \prime }$
E) $8 ^ { \prime } \times 96 ^ { \prime }$

Question 10

The path of a diver is given by $y = - \frac { 16 } { 25 } x ^ { 2 } + \frac { 24 } { 5 } x + 1$ where $y$ is the height (in feet) and $x$ is the horizontal distance from the end of the diving board (in feet). Find the maximum height of the diver.

A)13 feet
B)19 feet
C)10 feet
D)11 feet
E)16 feet

Question 11

Compare the graph of $p ( x ) = 4 ( x - 3 ) ^ { 2 } + 9$ with $p ( x ) = x ^ { 2 }$ .

A) $p ( x ) = 4 ( x - 3 ) ^ { 2 } + 9$ shifts right $3$ units, shifts downward $9$ units, and shrinks by a factor of $\frac { 1 } { 4 }$ .
B) $p ( x ) = 4 ( x - 3 ) ^ { 2 } + 9$ shifts right $3$ units, shifts upward $9$ units, and stretches by a factor of $4$ .
C) $p ( x ) = 4 ( x - 3 ) ^ { 2 } + 9$ shifts left $3$ units, shifts downward $9$ units, and stretches by a factor of $4$ .
D) $p ( x ) = 4 ( x - 3 ) ^ { 2 } + 9$ shifts right $3$ units, shifts upward $9$ units, and shrinks by a factor of $\frac { 1 } { 4 }$ .
E) $p ( x ) = 4 ( x - 3 ) ^ { 2 } + 9$ shifts left $3$ units, shifts upward $9$ units, and stretches by a factor of $4$ .

Question 12

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

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 13

Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } - 41 x ^ { 2 } + 400$ and determine the mutiplicity of each.

A) $x = 25$ , multiplicity 2; $x = 16$ , multiplicity 2
B) $x = 5$ , multiplicity 2; $x = - 4$ , multiplicity 2
C) $x = 25$ , multiplicity 2; $x = - 4$ , multiplicity 1
D) $x = - 5$ , multiplicity 2; $x = 4$ , multiplicity 2
E) $x = 5$ , multiplicity 1; $x = - 5$ , multiplicity 1; $x = - 4$ , multiplicity 1; $x = 4$ , multiplicity 1

Question 14

The sum of the length l and the width w of a rectangular region is 240 meters. Find the dimensions that produce the greatest area.

A) $l = 110 , w = 130$
B) $l = 125 ; \quad w = 115$
C) $l = 135 ; w = 105$
D) $l = 120 , \quad w = 120$
E) $l = 115 ; \quad w = 130$

Question 15

Describe the right-hand and the left-hand behavior of the graph of $u ( x ) = - \frac { 3 } { 7 } \left( x ^ { 3 } + 10 x ^ { 2 } - 3 x + 1 \right)$ .

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 negative, the graph rises to the left and falls to the right.
C)Because the degree is odd and the leading coefficient is negative, 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 negative, the graph rises to the left and falls to the right.

Question 16

Find an equation of the parabola below.

A) $y = - ( x - 3 ) ^ { 2 } + 2$
B) $y = - ( x - 2 ) ^ { 2 } + 3$
C) $y = - ( x + 3 ) ^ { 2 } - 2$
D) $y = ( x + 3 ) ^ { 2 } - 2$
E) $y = ( x - 3 ) ^ { 2 } - 2$

Question 17

The sum of the length l and the width w of a rectangular region is 210 meters. Write the area A as a function of w.

A) $A ( w ) = w ^ { 2 } - 210 w$
B) $A ( w ) = w ^ { 2 } + 210 w$
C) $A ( w ) = w - 210$
D) $A ( w ) = 210 - w$
E) $A ( w ) = 210 w - w ^ { 2 }$

Question 18

Find an equation of the parabola that has a vertex at $( - 5 , - 7 )$ and whose graph passes through the point $( - 6 , - 10 ).$

A) $y = 3 ( x + 7 ) ^ { 2 } - 5$
B) $y = - 3 ( x + 5 ) ^ { 2 } - 7$
C) $y = 3 ( x - 5 ) ^ { 2 } - 7$
D) $y = - 3 ( x + 5 ) ^ { 2 } + 7$
E) $y = 3 ( x + 5 ) ^ { 2 } + 7$

Question 19

Determine the vertex of the graph of the quadratic function $f ( x ) = x ^ { 2 } - 8$ .

A) $( 0,8 )$
B) $( - 8,0 )$
C) $( - 8 , - 8 )$
D) $( 0 , - 8 )$
E) $( 8,0 )$

Question 20

The fuel efficiency, in miles per gallon, for a certain midsize car at various speeds, in miles per hour, is given in the table below. $$\begin{array} { | c | c | c | c | } \hline \mathrm { mph } & \mathrm { mpg } & \mathrm { mph } & \mathrm { mpg } \\\hline 25 & 20 & 55 & 31 \\\hline 30 & 24 & 60 & 33 \\\hline 35 & 32 & 65 & 30 \\\hline 40 & 35 & 70 & 24 \\\hline 45 & 36 & 75 & 22 \\\hline 50 & 40 & & \\\hline\end{array}$$ Find a quadratic model for these data and use it to predict the fuel efficiency of this car when it is traveling at a speed of 40 mph. Do not round any values in your calculations but round the final answer to the nearest tenth.

A)33.7 mpg
B)35.7 mpg
C)32.7 mpg
D)33.9 mpg
E)34.9 mpg

Question 21

Write $f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 10 x - 29$ in the form $f ( x ) = ( x - k ) q ( x ) + r$ when $k = 6$ .

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

Question 22

Use synthetic division to divide. $\left( 5 x ^ { 3 } - 8 x ^ { 2 } - 7 x + 6 \right) \div ( x - 2 )$

A) $5 x ^ { 2 } + 3 x - 2$
B) $5 x ^ { 2 } + 2 x - 3$
C) $5 x ^ { 2 } - 5 x + 6$
D) $5 x ^ { 2 } - 13 x - 15$
E) $5 x ^ { 2 } + 2 x + 5$

Question 23

Use synthetic division to divide. $\left( 8 + 5 x ^ { 3 } + 46 x + 31 x ^ { 2 } \right) \div ( x + 2 )$

A) $5 x ^ { 2 } + 22 x + 8$
B) $5 x ^ { 2 } + 11 x + 5$
C) $5 x ^ { 2 } + 9 x + 2$
D) $5 x ^ { 2 } + 21 x + 4$
E) $5 x ^ { 2 } + 9 x + 20$

Question 24

Using the factors $( x - 5 )$ and $( x + 3 )$ , find the remaining factor(s) of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15$ and write the polynomial in fully factored form.

A) $f ( x ) = ( x - 5 ) ( x + 3 ) ( x - 1 )$
B) $f ( x ) = ( x - 5 ) ( x + 3 ) ^ { 2 }$
C) $f ( x ) = ( x - 5 ) ( x + 3 ) ( x + 1 )$
D) $f ( x ) = ( x - 5 ) ^ { 2 } ( x + 3 )$
E) $f ( x ) = ( x - 5 ) ( x + 3 ) ( x + 3 )$

Question 25

Use synthetic division to divide $x ^ { 3 } + 4 x ^ { 2 } + 4 x + 3$ by $x - 8$ .

A) $x ^ { 2 } + 4 x + 4 + \frac { 3 } { x - 8 }$
B) $x ^ { 2 } + 12 x + 100$
C) $x ^ { 2 } - 4 x - 92 - \frac { 797 } { x - 8 }$
D) $x ^ { 2 } + 4 x + 4$
E) $x ^ { 2 } + 12 x + 100 + \frac { 803 } { x - 8 }$

Question 26

Find the real zeros of the polynomial function $f ( x ) = x ^ { 5 } - 16 x ^ { 4 } + 64 x ^ { 3 }$ and determine the multiplicity of each.

A) $x = 0$ , multiplicity 2; $x = 8$ , multiplicity 3
B) $x = 0$ , multiplicity 2; $x = 8$ , multiplicity 1; $x = - 8$ , multiplicity 2
C) $x = 0$ , multiplicity 3; $x = 8$ , multiplicity 2
D) $x = 0$ , multiplicity 3; $x = - 8$ , multiplicity 2
E) $x = 0$ , multiplicity 2; $x = 8$ , multiplicity 2; $x = - 8$ , multiplicity 1

Question 27

Simplify the rational expression, $\frac { - 4 x ^ { 3 } + 10 x ^ { 2 } + 28 x - 16 } { - 4 x + 2 }$ , by using long division or synthetic division.

A) $x ^ { 2 } + 4 x + 8$
B) $x ^ { 2 } - 8 x + 4$
C) $x ^ { 2 } + 6 x - 8$
D) $x ^ { 2 } - 6 x + 16$
E) $x ^ { 2 } - 2 x - 8$

Question 28

Sketch the graph of the polynomial function below. $\begin{aligned}P ( x ) & = x ^ { 4 } - 10 x ^ { 3 } + 36 x ^ { 2 } - 54 x + 27 \\& = ( x - 3 ) ( x - 3 ) ( x - 3 ) ( x - 1 )\end{aligned}$

A)
B)
C)
D)
E)

Question 29

Find all the real zeros of the polynomial function $f ( x ) = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20$ and determine the multiplicity of each.

A) $x = 2$ , multiplicity 2; $x = - 5$ , multiplicity 1
B) $x = 2$ , multiplicity 1; $x = - 2$ , multiplicity 1; $x = - 5$ , multiplicity 1
C) $x = - 5$ , multiplicity 2; $x = - 2$ , multiplicity 1
D) $x = - 2$ , multiplicity 1; $x = 5$ , multiplicity 1; $x = - 5$ , multiplicity 1
E) $x = - 5$ , multiplicity 3

Question 30

Using the factors $( 3 x + 2 )$ and $( x - 1 )$ , find the remaining factor(s) of $f ( x ) = - 6 x ^ { 4 } + 23 x ^ { 3 } - 12 x ^ { 2 } - 11 x + 6$ and write the polynomial in fully factored form.

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

Question 31

Match the graph with one of the following functions below. Function I : $f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6$
Function II : $f ( x ) = x ^ { 3 } - x ^ { 2 } - 5 x - 3$
Function III : $f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 3 x - 18$
Function IV : $f ( x ) = x ^ { 3 } - x ^ { 2 } - 8 x + 12$
Function $\mathrm { V } : f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + 15 x + 9$

A)Function IV
B)Function II
C)Function III
D)Function V
E)Function I

Question 32

Sketch the graph of the polynomial function $P ( x ) = - x ^ { 3 } - x ^ { 2 } + 6 x$ . Do not use a graphing calculator.

A)
B)
C)
D)
E)

Question 33

Use synthetic division to divide. $\left( x ^ { 3 } - 48 x + 128 \right) \div ( x - 4 )$

A) $x ^ { 2 } + 4 x - 32$
B) $x ^ { 2 } - 4 x - 48$
C) $x ^ { 2 } + 8 x + 16$
D) $x ^ { 2 } + 12 x + 32$
E) $x ^ { 2 } + 16 x - 8$

Question 34

Use long division to divide. $\left( 2 x ^ { 2 } + 7 x + 3 \right) \div ( x + 3 )$

A) $2 x + 13 + \frac { 42 } { x + 3 }$
B) $2 x + 1$
C) $2 x + 13 + \frac { 14 } { x + 3 }$
D) $2 x + 14$
E) $- 2 x - 1$

Question 35

Use long division to divide. $\left( x ^ { 4 } - 2 x ^ { 2 } - 5 \right) \div \left( x ^ { 2 } - 3 x - 2 \right)$

A) $x ^ { 2 } + 3 x - 3$
B) $x ^ { 2 } - 3 x + 3$
C) $x ^ { 2 } + 3 x + 9 + \frac { 33 x + 13 } { x ^ { 2 } - 3 x - 2 }$
D) $x ^ { 2 } - 3 x + 3 + \frac { x - 2 } { x ^ { 2 } - 3 x - 2 }$
E) $x ^ { 2 } + 3 x - 3 - \frac { 4 } { x ^ { 2 } + 3 x - 3 }$

Question 36

Use synthetic division and the Factor Theorem to determine whether or not $x - 4$ is a factor of $P ( x ) = x ^ { 3 } + 2 x ^ { 2 } - x - 2$ .

A)Yes, $x - 4$ is a factor.
B)No, $x - 4$ is not a factor.

Question 37

Use long division to divide the first polynomial below by the second. $x ^ { 5 } + x ^ { 4 } + 7 x ^ { 3 } + x ^ { 2 } + 6 x - 27,$ $x ^ { 2 } + 6$

A) $x ^ { 3 } + x ^ { 2 } - 5 x + 3 + \frac { 1 } { x ^ { 2 } + 6 }$
B) $x ^ { 3 } + x ^ { 2 } + x - 5 + \frac { 3 } { x ^ { 2 } + 6 }$
C) $x ^ { 3 } + x ^ { 2 } + x + 3 + \frac { 1 } { x ^ { 2 } + 6 }$
D) $x ^ { 3 } + 3 x ^ { 2 } + x + 1 - \frac { 5 } { x ^ { 2 } + 6 }$
E) $x ^ { 3 } - 5 x ^ { 2 } + 3 x + 1 + \frac { 1 } { x ^ { 2 } + 6 }$

Question 38

Use long division to divide. $\left( x ^ { 3 } - 5 x ^ { 2 } + 9 x - 45 \right) \div ( x - 5 )$

A) $x ^ { 2 } + 15$
B) $x ^ { 2 } - 10 x + 53 - \frac { 247 } { x - 5 }$
C) $x ^ { 2 } - 10 x + 59 + \frac { 160 } { x - 5 }$
D) $x ^ { 2 } - 10 x + 53$
E) $x ^ { 2 } + 9$

Question 39

Use synthetic division and the Factor Theorem to determine whether or not $4 x + 3$ is a factor of $P ( x ) = - 12 x ^ { 3 } - 40 x ^ { 2 } + 52 x + 80$ .

A)No, $4 x + 3$ is not a factor.
B)Yes, $4 x + 3$ is a factor.

Question 40

A closed box is to be constructed from a rectangular sheet of cardboard that measures 25 inches by 12 inches. The box is made by cutting rectangles that measure x inches by 2x inches from two of the corners and by cutting two squares that measure x inches by x inches from the top and from the bottom of the rectangle, as shown in the following figure. Use a graphing calculator to determine what value of x (to the nearest thousandth of an inch) will produce a box with maximum volume?

A)2.296
B)9.186
C)1.814
D)7.260
E)1.614

Question 41

Find all the real zeros of $f ( x ) = 4 x ^ { 3 } + 28 x ^ { 2 } + 25 x + 6$ .

A) $x = - \frac { 1 } { 2 } , - 6$
B) $x = - \frac { 1 } { 3 } , \frac { 1 } { 6 }$
C) $x = - \frac { 1 } { 6 } , 2$
D) $x = - \frac { 1 } { 2 } , \frac { 1 } { 6 }$
E) $x = - \frac { 1 } { 2 } , 6$

Question 42

Find all the real zeros of $f ( x ) = 4 x ^ { 3 } - 5 x ^ { 2 } + 4 x - 5$ .

A) $x = \frac { 5 } { 4 } , - \frac { 4 } { 5 }$
B) $x = 4,1 , - 5$
C) $x = \pm 1 , \frac { 5 } { 4 }$
D) $x = \frac { 5 } { 4 }$
E) $x = \frac { 1 } { 4 } , - \frac { 1 } { 5 }$

Question 43

The length of a rectangular box is 1 inch more than twice the height of the box, and the width is 3 inches more than the height. If the volume of the box is 440 cubic inches, find the dimensions of the box.

A)5 in. by 11 in. by 8 in.
B)4 in. by 9 in. by 7 in.
C)109 in. by 219 in. by 112 in.
D)5.5 in. by 12 in. by 8.5 in.
E)106 in. by 213 in. by 109 in.

Question 44

Simplify the rational expression, $\frac { x ^ { 4 } - 5 x ^ { 3 } - 6 x ^ { 2 } + 32 x + 32 } { x ^ { 2 } - 8 x + 16 }$ , by using long division or synthetic division.

A) $x ^ { 2 } + 3 x + 2$
B) $- x ^ { 2 } - x + 2$
C) $x ^ { 2 } + x + 4$
D) $- x ^ { 2 } + 2 x + 4$
E) $x ^ { 2 } - 3 x - 4$

Question 45

Use the Rational Zero Test to list all possible rational zeros of the polynomial function $P ( x ) = 11 x ^ { 3 } - x ^ { 2 } + 8 x - 14$ .

A) $\pm 1 , \pm 11 , \pm 2 , \pm 7 , \pm 14$
B) $\pm 1 , \pm 11 , \pm \frac { 1 } { 14 } , \pm \frac { 11 } { 14 }$
C) $\pm 1 , \pm 11 , \pm \frac { 1 } { 2 } , \pm \frac { 11 } { 2 } , \pm \frac { 1 } { 7 } , \pm \frac { 11 } { 7 } , \pm \frac { 1 } { 14 } , \pm \frac { 11 } { 14 }$
D) $\pm 1 , \pm 14 , \pm \frac { 1 } { 11 } , \pm \frac { 14 } { 11 }$
E) $\pm 1 , \pm 2 , \pm 7 , \pm 14 , \pm \frac { 1 } { 11 } , \pm \frac { 2 } { 11 } , \pm \frac { 7 } { 11 } , \pm \frac { 14 } { 11 }$

Question 46

Find the zeros of the polynomial function below. If a zero is a multiple zero, state its multiplicity. $P ( x ) = x ^ { 3 } - 5 x ^ { 2 } - 15 x + 7$

A) $7,$ $- 1 - \sqrt { 2 },$ $- 1 + \sqrt { 2 }$
B) $7,$ $1 - \sqrt { 2 },$ $1 + \sqrt { 2 }$
C) $- 1,$ $7 - \sqrt { 2 },$ $7 + \sqrt { 2 }$
D) $1,$ $7 - \sqrt { 2 },$ $7 + \sqrt { 2 }$
E) $- 1,$ $- 7 - \sqrt { 2 },$ $- 7 + \sqrt { 2 }$

Question 47

Evaluate the power of i. $i ^ { 236 }$

A) $1$
B) $i$
C) $- 1$
D) $- i$
E) $0$

Question 48

Use the Rational Zero Test to list possible rational zeros for the polynomial function below. $P ( x ) = x ^ { 3 } - 7 x ^ { 2 } - 6 x + 20$

A) $\pm 1 , \pm 2 , \pm 5 , \pm 10 , \pm 20$
B) $\pm 1 , \pm 2 , \pm 4 , \pm 5 , \pm 10 , \pm 20$
C) $\pm 1 , \pm 2 , \pm 4 , \pm 5 , \pm 10$
D) $\pm 1 , \pm 2 , \pm 4 , \pm 10 , \pm 20$
E) $\pm 2 , \pm 4 , \pm 10 , \pm 20$

Question 49

An open box is to be made from a square piece of cardboard, 36 inches on a side, by cutting equal squares with sides of length x from the corners and turning up the sides (see figure below). After determining the function V, in terms of x, that represents the volume of the box, use a graphing utility to estimate the dimensions that will maximize its volume.

A) $12 ^ {||}$ × $12 ^ { ||}$ × $6^{||}$
B) $24^{||}$ × $24^{||}$ × $6^{||}$
C) $24^{||}$ × $24^{||}$ × $12 ^ { || }$
D) $18^ { ||}$ × $18^ { ||}$ × $12^ { ||}$
E) $6^ { ||}$ × $6^ { ||}$ × $3^ { ||}$

Question 50

Simplify and write the following complex number in standard form. $( - 6 - i ) + ( 5 - 10 i )$

A) $- 1 - 11 i$
B) $- 16 + 4 i$
C) $- 1 + 9 i$
D) $- 7 - 5 i$
E) $- 11 - 11 i$

Question 51

Find all real solutions of the polynomial equation $6 x ^ { 6 } - x ^ { 5 } + 29 x ^ { 4 } - 5 x ^ { 3 } + 19 x ^ { 2 } - 4 x - 4 = 0$ .

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

Question 52

Use the zero or root feature of a graphing utility to approximate the real zeros of the function $f ( x ) = x ^ { 6 } - 4 x ^ { 4 } + x ^ { 2 } + 6$ accurate to the nearest thousandth.

A) $\pm 1$ , 1.732, 1.414
B)-1.732, -1.414
C) $\pm$ 1.732, $\pm$ 1.414
D)1.732, 1.414
E) $\pm 1$ , - 1.732, 1.414

Question 53

Write the complex number $12 + \sqrt { - 16 }$ in standard form.

A) $12 + 4 i$
B) $- 4 i$
C) $12 + 16 i$
D) $28 i$
E) $12 - 4 i$

Question 54

Find real numbers a and b such that the equation $a + b i = 3 + 8 i$ is true.

A) $a = - 3 , b = - 8$
B) $a = 3 , b = - 8$
C) $a = - 3 , b = 8$
D) $a = 3 , b = 8$
E) $a = - 5 , b = 11$

Question 55

A farmer wants to enlarge an existing rectangular pig pen such that the total area of the new pen is twice as large as the original pen. The current pen's dimensions are 21' × 32'. The farmer wants to add 9 feet more to the long side than to the the short side. Find the dimensions of the new pen.

A)7 feet × 16 feet
B)28 feet × 48 feet
C)28 feet × 39 feet
D)28 feet × 37 feet
E)28 feet × 41 feet

Question 56

Find all real solutions of the polynomial equation $x ^ { 4 } - 8 x ^ { 3 } + 56 x - 49 = 0$ .

A) $x = 1,7 , \pm \sqrt { 7 }$
B) $x = 1,49$
C) $x = 1 , - 8 , - 7$
D) $x = 1 , - 49,14$
E) $x = 1 , \pm \sqrt { 7 }$

Question 57

A company that produces video games estimates that the profit $P$ (in dollars) for selling a new game is given by $P = - 82 x ^ { 3 } + 7250 x ^ { 2 } - 450,000,$ $0 \leq x \leq 80$ where $x$ is the advertising expense (in tens of thousands of dollars). Using this model, how much should the company spend on advertising to obtain a profit of $6,100,000? Round to the nearest dollar.

A)$368,914 or $764,017
B)$410,790 or $737,145
C)$344,417 or $778,509
D)$433,844 or $721,190
E)$315,902 or $794,255

Question 58

Find all the rational zeros of the function $f ( x ) = 2 x ^ { 5 } + x ^ { 4 } - 11 x ^ { 3 } + 7 x ^ { 2 } - 13 x + 6$ .

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) $x = \frac { 1 } { 2 } , - 3,2$

Question 59

Write the complex number in standard form. $- \sqrt { - 9 } - \sqrt { 64 }$

A) $- 8 + 3 i$
B) $- 8 - 3 i$
C) $8 - 3 i$
D) $- 3 - 8 i$
E) $3 - 8 i$

Question 60

Find all the rational zeros of the function $f ( x ) = - 3 x ^ { 4 } + 4 x ^ { 3 } + 52 x ^ { 2 } - 64 x - 64$ .

A) $x = - 2 , - 4,4 , \frac { 3 } { 2 }$
B) $x = - 3,8 , - 4$
C) $x = 2 , - 4,4 , - \frac { 2 } { 3 }$
D) $x = \frac { 1 } { 2 } , \frac { 4 } { 3 } , - \frac { 2 } { 3 } , 4$
E) $x = - 3,8 , \frac { 4 } { 3 } , - \frac { 2 } { 3 }$

Question 61

Simplify $( 2 + i ) ( - 9 + 7 i )$ and write the answer in standard form.

A) $23 + 5 i$
B) $- 25 - 61 i$
C) $- 65 + 5 i$
D) $- 65 - 11 i$
E) $- 25 + 5 i$

Question 62

Write $f ( x ) = x ^ { 5 } - 6 x ^ { 4 } + 15 x ^ { 3 } - 18 x ^ { 2 } + 10 x$ as a product of linear factors given that $2 - i$ and $1 - i$ are zeros of the polynomial.

A) $( x - 2 + i ) ^ { 3 } ( x - 1 - i ) ^ { 2 }$
B) $x ^ { 2 } ( x - 2 - i ) ( x - 1 + i ) ( x - 1 - i )$
C) $x ( x - 2 + i ) ( x - 2 - i ) ( x - 1 + i ) ( x - 1 - i )$
D) $x ^ { 3 } ( x - 2 - i ) ( x - 1 - i )$
E) $( x - 2 - i ) ^ { 2 } ( x - 1 - i ) ^ { 3 }$

Question 63

Simplify $\frac { 4 + 3 i } { 5 + 6 i }$ and write the answer in standard form.

A) $- \frac { 38 } { 61 } - \frac { 9 } { 61 } i$
B) $\frac { 38 } { 61 } + \frac { 9 } { 61 } i$
C) $\frac { 38 } { 61 } - \frac { 9 } { 61 } i$
D) $- \frac { 9 } { 61 } + \frac { 38 } { 61 } i$
E) $- \frac { 9 } { 61 } - \frac { 38 } { 61 } i$

Question 64

Write $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 25 x - 75$ as a product of linear factors.

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

Question 65

Use the quadratic formula to solve $x ^ { 2 } - 12 x + 205 = 0$ .

A) $x = - 133 \pm 13 i$
B) $x = - 7 \pm 13 i$
C) $x = 7 \pm 13 i$
D) $x = - 6 \pm 13 i$
E) $x = 6 \pm 13 i$

Question 66

Write the following expression as a complex number in standard form. $\frac { - 4 - 7 i } { - 5 - 8 i }$

A) $1 + i$
B) $\frac { 76 } { 89 } + \frac { 3 } { 89 } i$
C) $\frac { 32 } { 89 } + \frac { 35 } { 89 } i$
D) $\frac { 36 } { 89 } - \frac { 67 } { 89 } i$
E) $- \frac { 32 } { 39 } - \frac { 35 } { 39 } i$

Question 67

Plot the complex number below. $- 3 - 2 i$

A)
B)
C)
D)
E)

Question 68

Combine $( 9 + \sqrt { - 3 } ) ( 8 - \sqrt { - 21 } )$ and write the answer in standard form.

A) $72 + 3 \sqrt { 7 } i$
B) $( 72 + 3 \sqrt { 7 } ) + ( 8 \sqrt { 3 } - 9 \sqrt { 21 } ) i$
C) $8 \sqrt { 3 } - 3 \sqrt { 7 } i$
D) $( 8 \sqrt { 3 } + 3 \sqrt { 7 } ) + ( 72 - 9 \sqrt { 7 } ) i$
E) $3 \sqrt { 7 } - 8 \sqrt { 3 } i$

Question 69

Simplify and write the following complex number in standard form. $( - 6 - 5 i ) ( - 3 + 2 i )$

A) $3 i$
B) $18 - 10 i$
C) $- 6 + 17 i$
D) $28 + 3 i$
E) $- 9 - 3 i$

Question 70

Simplify $\frac { - 4 - 4 i } { 5 i }$ and write the answer in standard form.

A) $\frac { 4 } { 5 } + \frac { 4 i } { 5 }$
B) $- \frac { 4 } { 5 } + \frac { 4 i } { 5 }$
C) $- \frac { 4 } { 5 } - \frac { 4 i } { 5 }$ .
D) $- \frac { 4 } { 5 } - \frac { 4 i } { 5 }$
E) $\frac { 4 } { 5 } - \frac { 4 i } { 5 }$ .

Question 71

The demand and cost equations for a stethoscope are given by $p = 134 - 0.0002 x$
and
$C = 50 x + 160,000$ where $p$ is the unit price (in dollars), $C$ is the total cost (in dollars), and $x$ is the number of units. The total profit $P$ (in dollars) obtained by producing and selling $x$ units is given by $P = R - C = x p - C.$ Determine a price $p$ that would yield a profit of $6.9 million.

A)$36.62
B)$48.83
C)$97.65
D)$109.86
E)$73.24

Question 72

Determine the number of zeros of the polynomial function below. $f ( x ) = 9 + x ^ { 2 } - 7 x ^ { 5 } + 7 x ^ { 7 }$

A)6
B)8
C)3
D)7
E)5

Question 73

Simplify $( 5 + 2 i ) ^ { 2 } - ( 5 - 2 i ) ^ { 2 }$ and write the answer in standard form.

A) $0$
B) $40 i$
C) $50 + 40 i$
D) $50 + 8 i$
E) $10 + 8 i$

Question 74

Write the polynomial $x ^ { 4 } - 7 x ^ { 2 } - 8$ as the product of factors that are irreducible over the rationals.

A) $\left( x ^ { 2 } - 8 \right) \left( x ^ { 2 } - 1 \right)$
B) $\left( x ^ { 2 } - 8 \right) \left( x ^ { 2 } + 1 \right)$
C) $\left( x ^ { 2 } + 8 \right) \left( x ^ { 2 } + 1 \right)$
D) $\left( x ^ { 2 } + 8 \right) \left( x ^ { 2 } - 1 \right)$
E) $\left( x ^ { 2 } - 7 \right) \left( x ^ { 2 } - 1 \right)$

Question 75

Write $f ( x ) = x ^ { 4 } - 2 x ^ { 3 } - x ^ { 2 } - 38 x + 130$ as a product of linear factors given that $3 - i$ and $- 2 - 3 i$ are zeros of the polynomial.

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

Question 76

Find a polynomial function with real coefficients that the given zeros. $- 5 , - 5,1 + \sqrt { 3 } i$

A)Answers will vary. Sample answer: $f ( x ) = x ^ { 4 } + 8 x ^ { 3 } + 9 x ^ { 2 } - 10 x + 100$
B)Answers will vary. Sample answer: $f ( x ) = x ^ { 4 } + 8 x ^ { 3 } + 9 x ^ { 2 } - 10 x - 100$
C)Answers will vary. Sample answer: $f ( x ) = x ^ { 4 } - 8 x ^ { 3 } + 9 x ^ { 2 } - 10 x + 100$
D)Answers will vary. Sample answer: $f ( x ) = x ^ { 4 } + 8 x ^ { 3 } - 9 x ^ { 2 } - 10 x + 100$
E)Answers will vary. Sample answer: $f ( x ) = x ^ { 4 } + 8 x ^ { 3 } + 9 x ^ { 2 } + 10 x + 100$

Question 77

Classify whether the following numbers are in the Mandelbrot Set. I : $- 1$ II : $- i$ III : $3 i$

A)Only numbers I and III are in the Mandelbrot Set.
B)Only number II is in the Mandelbrot Set.
C)Only numbers II and III are in the Mandelbrot Set.
D)Only numbers I and II are in the Mandelbrot Set.
E)None are in the Mandelbrot Set.

Question 78

Simplify and write the following complex number in standard form. $\sqrt { - 684 } \cdot \sqrt { - 19 }$

A) $114 i$
B) $25 i$
C) $- 114$
D) $13 i$
E) $114$

Question 79

Simplify and write the following complex number in standard form. $( 5 + i ) ( 5 - i )$

A) $26$
B) $10$
C) $10 i$
D) $25 - i$
E)0

Question 80

Find all the zeros of the polynomial function $P ( x ) = x ^ { 3 } + 10 x ^ { 2 } + 34 x + 40$ and write the polynomial as a product of its linear factors.

A) $4$ , $3 + i$ , $3 + i$ ; $P ( x ) = ( x + 4 ) ( x + 3 + i ) ( x + 3 + i )$
B) $- 4$ , $- 3 + i$ , $- 3 - i$ ; $P ( x ) = ( x - 4 ) ( x - 3 + i ) ( x - 3 - i )$
C) $4$ , $3 - i$ , $3 + i$ ; $P ( x ) = ( x + 4 ) ( x - 3 + i ) ( x - 3 - i )$
D) $- 4$ , $- 3 - i$ , $- 3 + i$ ; $P ( x ) = ( x + 4 ) ( x + 3 + i ) ( x + 3 - i )$
E) $- 4$ , $- 3 - i$ , $- 3 - i$ ; $P ( x ) = ( x - 4 ) ( x - 3 - i ) ( x - 3 - i )$