Deck 3: Polynomial and Rational Functions

Question 1

Use synthetic division and the Factor Theorem to determine whether or not is a factor of .

A)Yes, is a factor.
B)No, is not a factor.

Answer 1

No, is not a factor.

Question 2

Find the real zeros of the polynomial function below by factoring.

A)
B)
C)
D)
E)

Answer 2

Question 3

Write as the product of and its reduced polynomial.

A)
B)
C)
D)
E)

Answer 3

Question 4

Use synthetic division and the Factor Theorem to determine whether or not is a factor of .

A)Yes, a factor.
B)No, a factor.

Question 5

Examine the leading term and determine the far-left and far-right behavior of the graph of the polynomial function .

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

Question 6

Graph the polynomial . Estimate, to the nearest tenth, the coordinates of the points where has a relative maximum or a relative minimum. For each point, indicate whether the y value is a relative maximum or a relative minimum.

A) has a relative maximum of 6.0 at and a relative minimum of at .
B) has a relative maximum of 6.0 at and a relative minimum of at .
C) has a relative maximum of 6.0 at and a relative minimum of 108.0 at .
D) has a relative maximum of 6.0 at and a relative minimum of at .
E) has a relative maximum of 6.0 at and a relative minimum of at .

Question 7

Use long division to divide the first polynomial below by the second.

A)
B)
C)
D)
E)

Question 8

Use synthetic division to determine which of the following values of c is a zero of .

A)-1
B)-5
C)-7
D)-6
E)-3

Question 9

Use synthetic division to determine which of the following values of c is a zero of . 7, 7, 4, 3, 6

A)-3
B)-6
C)7
D)4
E)-7

Question 10

Use the Remainder Theorem to find for and .

A)-2
B)0
C)10
D)-10
E)-12

Question 11

Use synthetic division to divide by .

A)
B)
C)
D)
E)

Question 12

Find the real zeros of the polynomial function by factoring.

A)1, -2, -24
B)1, -6, 4
C)0, -12, -2
D)0, 6, -4
E)0, 12, 2

Question 13

Sketch the graph of the polynomial function below.

A)
B)
C)
D)
E)

Question 14

Use the Remainder Theorem to find P( 8)for .

A)4,368
B)-4,352
C)-4,864
D)374,288
E)4880

Question 15

A rectangular box has a volume of cubic inches. The height of the box is inches. The width of the box is inches. Find the length of the box in terms of x .

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

Question 16

Sketch the graph of the polynomial function . Do not use a graphing calculator.

A)
B)
C)
D)
E)

Question 17

Determine the value of k so that the divisor is a factor of the dividend.

A)24
B)48
C)18
D)-24
E)-48

Question 18

Determine the x -intercepts of the graph of the polynomial below. For each x -intercept, use the Even and Odd Powers of Theorem to determine whether the graph of the polynomial crosses the x -axis or intersects but does not cross the x -axis.

A)crosses the x -axis at , , and
B)crosses the x -axis at ; intersects the x -axis at and
C)crosses the x -axis at and ; intersects the x -axis at
D)crosses the x -axis at and ; intersects the x -axis at
E)crosses the x -axis at and ; intersects the x -axis at

Question 19

The number of ways you can select three cards from a stack of n cards, in which the order of selection is important, is given by . Use the Remainder Theorem to determine the number of ways you can select three cards from a stack of 9 cards.

A)504 ways
B)336 ways
C)210 ways
D)120 ways
E)24 ways

Question 20

Determine the x -intercepts of the graph of . For each x -intercept, use the Even and Odd Powers of Theorem to determine whether the graph of P crosses the x -axis or intersects but does not cross the x -axis.

A)crosses the x -axis at and
B)crosses the x -axis at ; intersects but does not cross at
C)crosses the x -axis at ; intersects but does not cross at
D)crosses the x -axis at ; intersects but does not cross at
E)crosses the x -axis at ; intersects but does not cross at

Question 21

Find the smallest positive integer and the largest negative integer that, by the Upper- and Lower-Bound Theorem, are upper and lower bounds, respectively, for the real zeros of the polynomial function .

A)upper bound 3, lower bound -4
B)upper bound 20, lower bound -28
C)upper bound 28, lower bound -28
D)upper bound 4, lower bound -1
E)upper bound 20, lower bound -20

Question 22

Find the smallest positive integer and the largest negative integer that, by the Upper- and Lower-Bound Theorem, are upper and lower bounds for the real zeros of the polynomial function below.

A)upper bound 2, lower bound -5
B)upper bound 3, lower bound -5
C)upper bound 2, lower bound -1
D)upper bound 1, lower bound -1
E)upper bound 3, lower bound -1

Question 23

Find the zeros of the polynomial function below. If a zero is a multiple zero, state its multiplicity.

A)3, 4 (multiplicity 2)
B)-3, 4 (multiplicity 2)
C)3, -4 (multiplicity 2)
D)-4, -3 (multiplicity 2)
E)4, 3 (multiplicity 2)

Question 24

Find all of the zeros of the polynomial function and state the multiplicity of each zero.

A)16, multiplicity of 2 6, multiplicity of 1
B)-4, multiplicity of 2 4, multiplicity of 2 -6, multiplicity of 1
C)-4, multiplicity of 1 4, multiplicity of 1 6, multiplicity of 3
D)-16, multiplicity of 1 -6, multiplicity of 3
E)4, multiplicity of 2 -6, multiplicity of 1

Question 25

A company constructs boxes from rectangular pieces of cardboard that measure 14 inches by 10 inches. An open box is formed by cutting squares that measure x inches by x inches from each corner of the cardboard and folding up the sides. Determine (to the nearest tenth of an inch)the x value that maximizes the volume of the box.

A) inches
B)2.2 inches
C)2.7 inches
D)3.8 inches
E)1.9 inches

Question 26

A cube measures n inches on each edge. If a slice 5 inches thick is cut from one face of the cube, the resulting solid has a volume of 2250 cubic inches. Find n .

A)16 inches
B)12 inches
C)15 inches
D)8 inches
E)14 inches

Question 27

Use translation and/or reflection concepts to explain how the graph of P can be used to produce the graph of Q .

A)Shift the graph of P horizontally 6 units to the left.
B)Shift the graph of P vertically downward 6 units ..
C)Shift the graph of P horizontally 6 units to the right.
D)Shift the graph of P vertically upward 6 units .

Question 28

Use Descartes Rule of Signs to state the number of possible positive and negative real zeros of the polynomial function below.

A)three or one positive zeros; three or one negative zeros
B)two or no positive zeros; four, two, or no negative zeros
C)six, four, two, or no positive zeros; no negative zeros
D)no positive zeros; six, four, two, or no negative zeros
E)four, two, or no positive zeros; two or no negative zeros

Question 29

Find the zeros of the polynomial function below and state the multiplicity of each zero.

A)7 (multiplicity 1), -4 (multiplicity 2)
B)7 (multiplicity 2), -4 (multiplicity 1)
C)-7 (multiplicity 1), 4 (multiplicity 2)
D)-7 (multiplicity 2), 4 (multiplicity 1)
E)7 (multiplicity 2), 4 (multiplicity 1)

Question 30

A company constructs boxes from rectangular pieces of cardboard that measure 10 inches by 11 inches. An open box is formed by cutting squares that measure x inches by x inches from each corner of the cardboard and folding up the sides. Express the volume V of the box as a function of x .

A)
B)
C)
D)
E)

Question 31

A software company produces a computer game. The company has determined that its profit P , in dollars, from the manufacture and sale of x games is given by where . How many games, to the nearest unit, does the company need to produce and sell to obtain the maximum profit?

A)5711 games
B)5756 games
C)5716 games
D)5706 games
E)5701 games

Question 32

Use Descartes Rule of Signs to determine the number of possible positive real zeros and the number of possible negative real zeros of .

A)positive zeros: 2 or 0 negative zeros: 3 or 1
B)positive zeros: 1 negative zeros: 2 or 0
C)positive zeros: 3 or 1 negative zeros: 2 or 0
D)positive zeros: 2 negative zeros: 1
E)positive zeros: 2 or 0 negative zeros: 1

Question 33

A software company produces a computer game. The company has determined that its profit P , in dollars, from the manufacture and sale of x games is given by where . What is the maximum profit, to the nearest thousand dollars, the company can expect from the sale of its games?

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

Question 34

Use the Rational Zero Theorem to list all possible rational zeros of the polynomial function .

A)
B)
C)
D)
E)

Question 35

A closed box is to be constructed from a rectangular sheet of cardboard that measures 21 inches by 10 inches. The box is made by cutting rectangles that measure x inches by 2 x 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)1.918
B)7.709
C)1.352
D)1.513
E)6.082

Question 36

Use the Rational Zero Theorem to list possible rational zeros for the polynomial function below.

A)
B)
C)
D)
E)

Question 37

Use translation and/or reflection concepts to explain how the graph of P can be used to produce the graph of Q .

A)Shift the graph of P unit to the left.
B)Shift the graph of P unit .
C)Shift the graph of P unit .
D)Shift the graph of P horizontally unit to the right.

Question 38

Find the zeros of the polynomial function . If a zero is a multiple zero, state its multiplicity.

A)-1, -12, 2
B)0, -6, 4
C)-3, 2, 4
D)3, -2, -4
E)1, -6, -4

Question 39

Find the zeros of the polynomial function below. If a zero is a multiple zero, state its multiplicity.

A)1, ,
B) , ,
C) , ,
D) , ,
E)1, ,

Question 40

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 1496 cubic inches, find the dimensions of the box.

A)373 in. by 747 in. by 376 in.
B)7 in. by 15 in. by 10 in.
C)8.5 in. by 18 in. by 11.5 in.
D)8 in. by 17 in. by 11 in.
E)370 in. by 741 in. by 373 in.

Question 41

Find a polynomial function of lowest degree with integer coefficients, whose leading coefficient is equal to 3, and that has the zeros , and .

A)
B)
C)
D)
E)

Question 42

Find a polynomial function of degree 4 with integer coefficients, whose leading coefficient is equal to one, and that has the zeros and .

A)
B)
C)
D)
E)

Question 43

Find a polynomial function P with real coefficients that is degree 3, has zeros and and .

A)
B)
C)
D)
E)

Question 44

Given is a zero of the polynomial function , find the remaining zeros of the polynomial.

A)
B)
C)
D)
E)5 i, -5, 4 + i , 4 - i

Question 45

Determine the domain of the rational function below.

A)all real numbers except 5
B)all real numbers except -4 and 4
C)all real numbers except 4
D)all real numbers
E)all real numbers except -5 and 5

Question 46

A propane tank has the shape of a circular cylinder with a hemisphere at each end. The cylinder is 6 feet long and the volume of the tank is cubic feet. Find, to the nearest thousandth of a foot, the length of the radius x.

A) ft
B) ft
C) ft
D)1.041 ft
E) ft

Question 47

Use the given zero to find the remaining zeros of the polynomial function below.

A)
B)
C)
D)
E)

Question 48

Find all the zeros of the polynomial function and write the polynomial as a product of its leading coefficient and its linear factors.

A)
B)
C)
D)
E)

Question 49

Find all vertical asymptotes of the rational function .

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

Question 50

Find a polynomial function P of degree 3 with integer coefficients that has zeros and .

A)
B)
C)
D)
E)

Question 51

The number of ways one can select three cards from a group of n cards (assuming the order of the selection matters), where , is given by . For a certain card trick a magician has determined that there are exactly 5814 ways to choose three cards from a given group. How many cards are in the group?

A)18 cards
B)17 cards
C)19 cards
D)20 cards
E)21 cards

Question 52

Find a polynomial function of lowest degree with integer coefficients that has the zeros 8, -1, -2.

A)
B)
C)
D)
E)

Question 53

Given is a zero of the polynomial function , find the remaining zeros of the polynomial.

A)
B)
C)
D)2, -1 + 5 i
E)

Question 54

Find all the zeros of the polynomial function and write the polynomial as a product of its linear factors.

A)5, 2 + 2 i, 2 + 2 i;
B)5, , ;
C)5, 2 2 i, 2 + 2 i ;
D)5, 2 2 i, 2 2 i ;
E)5, 2 2 i, 2 + 2 i ;

Question 55

Use a graph and your knowledge of the zeros of polynomial functions to determine the exact values of all the solutions of the equation .

A)
B)
C)
D)
E)

Question 56

Find a polynomial function P of degree 5 with integer coefficients that has zeros , (multiplicity 2), , and .

A)
B)
C)
D)
E)

Question 57

Find a polynomial function of lowest degree with integer coefficients, whose leading coefficient is one, and that has the zeros 8, , and 3 i .

A)
B)
C)
D)
E)

Question 58

Find the horizontal asymptote of the rational function .

A)
B)
C)
D)
E)

Question 59

Find all vertical asymptotes of the rational function below.

A)
B)
C)
D)
E)

Question 60

Use a graph and your knowledge of the zeros of polynomial functions to determine the exact values of all the solutions of the equation below.

A) (multiplicity 2)
B) (multiplicity 2)
C) (multiplicity 2)
D) (multiplicity 2)
E) (multiplicity 2)

Question 61

A cylindrical soft drink can is to be made so that it will have a volume of 203 milliliters. If r is the radius of the can in centimeters, then the total surface area A of the can is given by the rational function . Does the graph of A have a slant asymptote?

Question 62

Determine the vertical and slant asymptotes of the rational function below.

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

Question 63

The cost, in dollars, of producing x golf balls is given by . The average cost per golf ball is given by . Find the average cost of producing golf balls.

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

Question 64

Determine the vertical and horizontal asymptotes of the rational function below.

A)
B)
C)
D)
E)

Question 65

Find the horizontal asymptote of the rational function .

A)
B)
C)
D)
E)

Question 66

The rational function models the number of milligrams of medication in the bloodstream of a patient t hours after 350 milligrams of the medication have been injected into the patient s bloodstream. What will M approach as ?

A)12 mg
B)16 mg
C)8 mg
D)50 mg
E)0 mg

Question 67

A music company expects that the monthly sales S , in thousands, of a new music CD it has produced will be closely approximated by where t is the number of months after the CD is released. Find the monthly sales the company expects months after the CD is released. Round to the nearest 100 CDs.

A)3700 CDs
B)3500 CDs
C)3400 CDs
D)4300 CDs
E)2900 CDs

Question 68

The cost, in dollars, of producing x cellular telephones is given by . The average cost per telephone is . Find the average cost per telephone, rounded to the nearest cent, when telephones are produced.

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

Question 69

The cost, in dollars, of producing x cellular telephones is given by . The average cost per telephone is . Use a graphing calculator to determine the minimum average cost per telephone, rounded to the nearest cent.

A)$4.61
B)$37.00
C)$28.00
D)$20.08
E)$109.00

Question 70

Determine the intercepts, vertical and horizontal asymptotes and sketch the graph of the rational function .

A)
B)
C)
D)
E)

Question 71

The rational function models the number of milligrams of medication in the bloodstream of a patient t hours after 600 milligrams of the medication have been injected into the patient s bloodstream. Find rounded to the nearest milligram.

A)210 mg
B)180 mg
C)60 mg
D)10 mg
E)108 mg

Question 72

Find the slant asymptote of the rational function .

A)
B)
C)
D)
E)

Question 73

Sketch the graph of the rational function .

A)
B)
C)
D)
E)

Question 74

Determine the vertical and slant asymptotes of the rational function below.

A)
B)
C)
D)
E)

Question 75

A cylindrical soft drink can is to be made so that it will have a volume of 365 milliliters. If r is the radius of the can in centimeters, then the total surface area A of the can is given by the rational function . Use the graph of A to estimate the value of r , to the nearest tenth of a centimeter, that will yield the minimum total surface area?

A)7.1 cm
B)9.7 cm
C)19.4 cm
D)3.9 cm
E)3.1 cm

Question 76

A music company expects that the monthly sales S , in thousands, of a new music CD it has produced will be closely approximated by where t is the number of months after the CD is released. Use S to predict the month in which sales are expected to reach a maximum.

A)Sales will reach a maximum in the fifth month.
B)Sales will reach a maximum in the tenth month.
C)Sales will reach a maximum in the third month.
D)Sales will reach a maximum in the eleventh month.
E)Sales will reach a maximum in the sixth month.

Question 77

The cost, in dollars, of producing x golf balls is given by . The average cost per golf ball is given by . Find the average cost per ball as the number of golf balls produced increases.

A)
B)
C)
D)
E)

Question 78

Determine the vertical and horizontal asymptotes of the rational function below.

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

Question 79

Determine the vertical and slant asymptotes of the rational function below.

A)
B)
C)
D)
E)