Deck 4: Polynomial Functions

Question 1

Determine the vertex of the graph of the quadratic function .

A)
B)
C)
D)
E)

Answer 1

Question 2

From the graph of the quadratic function , determine the equation of the axis of symmetry.

A)
B)
C)
D)
E)

Answer 2

Question 3

Write the quadratic function in standard form.

A)
B)
C)
D)
E)

Answer 3

Question 4

Write the standard form of the equation of the parabola that has a vertex at and passes through the point .

A)
B)
C)
D)
E)

Question 5

Write the standard form of the equation of the parabola that has a vertex at and passes through the point .

A)
B)
C)
D)
E)

Question 6

Find two positive real numbers whose product is a maximum and whose sum of the first number and four times the second is 280.

A)
B)
C)
D)
E)

Question 7

Compare the graph of with .

A) shifts left 7 units, shifts downward 5 units, and stretches by a factor of 9.
B) shifts left 7 units, shifts upward 5 units, and stretches by a factor of 9.
C) shifts right 7 units, shifts downward 5 units, and shrinks by a factor of
)
D) shifts right 7 units, shifts upward 5 units, and shrinks by a factor of
)
E) shifts right 7 units, shifts upward 5 units, and stretches by a factor of 9.

Question 8

Describe the right-hand and the left-hand behavior of the graph of .

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

Question 9

Find two positive real numbers whose product is a maximum and whose sum is 156.

A)
B)
C)
D)
E)

Question 10

The height, , of a punted rugby ball is given by where x is the horizontal distance in feet from the point where the ball is punted. How far, horizontally, is the ball from the kicker when it is at its highest point?

A)
B)
C)
D)
E)

Question 11

From the graph of the quadratic function , determine the equation of the axis of symmetry.

A)
B)
C)
D)
E)

Question 12

A farmer has 336 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)
B)
C)
D)
E)

Question 13

Write the quadratic function in standard form.

A)
B)
C)
D)
E)

Question 14

Match the equation with its graph.

A)
B)
C)
D)
E)

Question 15

Compare the graph of with .

A) - 8 shifts left 4 units, shifts downward 8 units, and shrinks by a factor of
)
B) - 8 shifts right 4 units, shifts upward 8 units, and shrinks by a factor of
)
C) - 8 shifts left 16 units, shifts upward 8 units, and shrinks by a factor of
)
D) - 8 shifts right 16 units, shifts upward 8 units, and shrinks by a factor of
)
E) - 8 shifts right 4 units, shifts downward 8 units, and shrinks by a factor of
)

Question 16

Determine the x-intercept(s) of the quadratic function .

A)
B)
C)
D)
E) no x-intercept(s)

Question 17

Determine the x-intercept(s) of the quadratic function .

A)
B)
C) no x-intercept(s)
D)
E)

Question 18

Write the quadratic function in standard form.

A)
B)
C)
D)
E)

Question 19

A Norman window has the shape of a rectangle surmounted by a semicircle as in the figure below. If the perimeter of the window is , express the area, A, as a function of the width, x, of the window.

A)
B)
C)
D)
E)

Question 20

Find the standard form of the quadratic function shown below:

A)
B)
C)
D)
E)

Question 21

Use synthetic division to divide.

A)
B)
C)
D)
E)

Question 22

Use synthetic division to divide.

A)
B)
C)
D)
E)

Question 23

Use synthetic division to divide.

A)
B)
C)
D)
E)

Question 24

Use synthetic division to divide.

Question 25

Use long division to divide.

A)
B)
C)
D)
E)

Question 26

Using a graphing utility, graph and approximate the zeros and their multiplicity.

A)
B)
C)
D)
E)

Question 27

Write in the form when .

A)
B)
C)
D)
E)

Question 28

Find all real zeros of the polynomial and determine the multiplicity of each.

A)
B)
C)
D)
E)

Question 29

If , use synthetic division to evaluate .

A)
B)
C)
D)
E)

Question 30

An open box is to be made from a square piece of cardboard, 32 inches on a side, by cutting equal squares with sides of length x from the corners and turning up the sides (see figure below). If the volume of the box is represented by , determine the domain of .

A)
B)
C)
D)
E)

Question 31

Use synthetic division to divide.

A)
B)
C)
D)
E)

Question 32

Use long division to divide.

A)
B)
C)
D)
E)

Question 33

If is a root of , use synthetic division to factor the polynomial completely and list all real solutions of the equation.

A) ;
B) ;
C) ;
D) ;
E) ;

Question 34

Find all real zeros of the polynomial and determine the multiplicity of each.

A)
B)
C)
D)
E)

Question 35

Describe the right-hand and the left-hand behavior of the graph of .

A) Because the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right.
B) Because the degree is odd and the leading coefficient is positive, the graph rises to the left and rises to the right.
C) Because the degree is odd and the leading coefficient is positive, the graph falls to the left and falls to the right.
D) Because the degree is odd and the leading coefficient is positive, the graph rises to the left and falls 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 36

Find all real zeros of the polynomial and determine the multiplicity of each.

A)
B)
C)
D)
E)

Question 37

Describe the right-hand and the left-hand behavior of the graph of .

A) Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right.
B) Because the degree is odd and the leading coefficient is negative, the graph falls to the left and rises to the right.
C) Because the degree is odd and the leading coefficient is positive, the graph falls to the left and falls 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 38

An open box is to be made from a square piece of cardboard, 21 inches on a side, by cutting equal squares with sides of length x from the corners and turning up the sides (see figure below). Determine the function, V, in terms of x, that represents the volume of the box.

A)
B)
C)
D)
E)

Question 39

Write in the form when , and demonstrate that .

Question 40

Use long division to divide.

A)
B)
C)
D)
E)

Question 41

Find all zeros of the function .

A)
B)
C)
D)
E)

Question 42

Using the factors and , find the remaining factor(s) of and write the polynomial in fully factored form.

A)
B)
C)
D)
E)

Question 43

Write as a product of linear factors.

A)
B)
C)
D)
E)

Question 44

Find all zeros of the function .

A)
B)
C)
D)
E)

Question 45

Use the zero or root feature of a graphing utility to approximate the zeros of accurate to the nearest thousandth.

A)
B)
C)
D)
E)

Question 46

If is a root of , use synthetic division to factor the polynomial completely and list all real solutions of the equation.

A) ;
B) ;
C) ;
D) ;
E) ;

Question 47

Given is a root, determine all other roots of .

A)
B)
C)
D)
E)

Question 48

Use the zero or root feature of a graphing utility to approximate the zeros of the function accurate to the nearest thousandth.

A)
B)
C)
D)
E)

Question 49

Using the factors and , find the remaining factor(s) of and write the polynomial in fully factored form.

A)
B)
C)
D)
E)

Question 50

If is a root of , use synthetic division to factor the polynomial completely and list all real solutions of the equation.

A)
B)
C)
D)
E)

Question 51

Find all zeros of the function .

A)
B)
C)
D)
E)

Question 52

Given is a root, determine all other roots of .

A)
B)
C)
D)
E)

Question 53

Find all real solutions of the polynomial equation .

A)
B)
C)
D)
E)

Question 54

Find all real solutions of the polynomial equation .

A)
B)
C)
D)
E)

Question 55

Find all the rational zeros of the function .

A)
B)
C)
D)
E)

Question 56

Write as a product of linear factors.

A)
B)
C)
D)
E)

Question 57

Given is a root, determine all other roots of .

A)
B)
C)
D)
E)

Question 58

Simplify the rational expression, , by using long division or synthetic division.

A)
B)
C)
D)
E)

Question 59

Simplify the rational expression, , by using long division or synthetic division.

A)
B)
C)
D)
E)

Question 60

Find all the rational zeros of the function .

A)
B)
C)
D)
E)

Question 61

After determining whether the variation model below is of the form or , find the value of k.

A)
B)
C)
D)
E)

Question 62

Determine whether the variation model below is of the form or .

A)
B)

Question 63

After determining whether the variation model below is of the form or , find the value of k.

A)
B)
C)
D)
E)

Question 64

Find a mathematical model for the verbal statement: "z varies directly as the square of x and inversely as p."

A)
B)
C)
D)
E)

Question 65

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

A) 3 positive reals or 1 positive real; 3 negative reals or 1 negative real
B) no positive reals; no negative reals
C) 3 positive reals or 1 positive real; 1 negative real
D) 1 positive real; 3 negative reals or 1 negative real
E) 3 positive reals or 1 positive real; no negative reals

Question 66

Find all the real zeros of .

A)
B)
C)
D)
E)

Question 67

Assume that y is directly proportional to x. If and , determine a linear model that relates y and x.

A)
B)
C)
D)
E)

Question 68

Find a mathematical model for the verbal statement:

A)
B)
C)
D)
E)

Question 69

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

A) 1 positive real; 3 negative reals or 1 negative real
B) no positive reals; no negative reals
C) 1 positive real; 5 negative reals or 3 negative reals or 1 negative real
D) 5 positive reals or 3 positive reals or 1 positive real; no negative reals
E) 3 positive reals or 1 positive real; 3 negative reals or 1 negative real

Question 70

Hooke's law states that the magnitude of force, F, required to stretch a spring x units beyond its natural length is directly proportional to x. If a force of 2 pounds stretches a spring from its natural length of 10 inches to a length of 10.7 inches, what force will stretch the spring to a length of 11.5 inches? Round answer to nearest hundredth.

A)
B)
C)
D)
E)

Question 71

Find all the real zeros of .

A)
B)
C)
D)
E)

Question 72

The sales tax on an item with a retail price of $972 is $77.76. Create a variational model that gives the retail price, y, in terms of the sales tax, x, and use it to determine the retail price of an item that has a sales tax of $92.34.

A) $1156.24
B) $1125.88
C) $1154.25
D) $1166.48
E) $1098.56

Question 73

The electrical resistance, R, of a wire is directly proportional to its length, l, and inversely proportional to the square of its diameter, d. A wire 100 meters long of diameter 5 millimeters has a resistance of 8 ohms. Find the resistance of a wire made of the same material that has a diameter of 4 millimeters and is 64 meters long.

A)
B)
C)
D)
E)

Question 74

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

A) no positive reals; no negative reals
B) 1 positive real; 3 negative reals or 1 negative real
C) 0 positive real; 3 negative reals or 1 negative real
D) 3 positive reals or 1 positive real; no negative reals
E) 3 positive reals or 1 positive real; 3 negative reals or 1 negative real

Question 75

Determine whether the variation model below is of the form or .

A)
B)

Question 76

The simple interest on an investment is directly proportional to the amount of the investment. By investing $7750 in a certain certificate of deposit, you obtained an interest payment of $372.00 after 1 year. Determine a mathematical model that gives the interest, I , for this CD after 1 year in terms of the amount invested, P.

A)
B)
C)
D)
E)