Question 1

Find all values of   such that

Accepted Answer

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

Question 2

Express the statement as a formula that involves the variables u, v and a constant of proportionality k, and determine the value of k from the condition : u is directly proportional to v and if v = 40, then u = 3

Accepted Answer

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

Question 3

Applying the first theorem on bounds for real zeros of polynomials, determine the smallest and largest integers that are upper and lower bounds, respectively, for the real solutions of the equation. With the aid of a graphing utility, discuss the validity of the bounds.

Accepted Answer

A) The upper bound is   , the lower bound is
  . 
B) The upper bound is   , the lower bound is
  . 
C) The upper bound is   , the lower bound is
  . 
D) The upper bound is   , the lower bound is
  . 
E) The upper bound is   , the lower bound is
  . 
A) The upper bound is   , the lower bound is
  . 
B) The upper bound is   , the lower bound is
  . 
C) The upper bound is   , the lower bound is
  . 
D) The upper bound is   , the lower bound is
  . 
E) The upper bound is   , the lower bound is
  .

Question 4

Find the quotient and remainder if   is divided by

Accepted Answer

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

Question 5

Find all values of   such that

Accepted Answer

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

Question 6

Poiseuille's law states that the blood flow rate F ( in L/min ) through a major artery is directly proportional to the product of the fourth power of the radius r and the blood pressure P. During heavy exercise, normal blood flow rates sometimes triple. If the radius of a major artery increases by 7%, approximately how much harder must the heart pump?

Accepted Answer

A) about 1.31 times as hard 
B) about 3.43 times as hard 
C) about 2.29 times as hard 
D) about 2.04 times as hard 
E) about 1.92 times as hard 
A) about 1.31 times as hard 
B) about 3.43 times as hard 
C) about 2.29 times as hard 
D) about 2.04 times as hard 
E) about 1.92 times as hard

Question 7

The pressure P acting at a point in a liquid is directly proportional to the distance d from the surface of the liquid to the point. Express P as a function of d by means of a formula that involves a constant of proportionality k. In a certain oil tank, the pressure at a depth of 8 feet is 472. Find the value of k.

Accepted Answer

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

Question 8

A polynomial f ( x ) with real coefficients and leading coefficient 1 has the given zeros and degree. Express f ( x ) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R.

Accepted Answer

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

Question 9

Find all solutions of the equation.

Accepted Answer

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

Question 10

Find a polynomial with leading coefficient of   , degree   , and zeros: -   .

Accepted Answer

A) f ( x ) = x 4  - 3x 3  - 26x 2  + 26x + 160 
B) f ( x ) = x 4  - 6x 2  - 22x + 40 
C) f ( x ) = x 4  - 26x 2  + 48x + 160 
D) f ( x ) = x 4  - 3x 3  - 26x 2  + 160 
E) f ( x ) = x 4  - 3x 3  - 26x 2  + 48x + 160 
A) f ( x ) = x 4  - 3x 3  - 26x 2  + 26x + 160 
B) f ( x ) = x 4  - 6x 2  - 22x + 40 
C) f ( x ) = x 4  - 26x 2  + 48x + 160 
D) f ( x ) = x 4  - 3x 3  - 26x 2  + 160 
E) f ( x ) = x 4  - 3x 3  - 26x 2  + 48x + 160

Question 11

From a rectangular piece of cardboard having dimensions W = 30 and L = 32 an open box is to be made by cutting out identical squares of area x 2  from each corner and turning up the sides (see Illustration). Find all positive values of x such that the volume of the box V ( x ) > 0. Include only allowable values of x in your answer.

Accepted Answer

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

Question 12

Use the factor theorem to decide whether   is a factor of the polynomial.   ;

Accepted Answer

A) x - c is a factor 
B) x - c is not a factor 
A) x - c is a factor 
B) x - c is not a factor

Question 13

Use synthetic division to decide whether   is a zero of   .   ;

Accepted Answer

A) x - c is not a zero 
B) x - c is a zero 
A) x - c is not a zero 
B) x - c is a zero

Question 14

A meteorologist determines that the temperature T   for a certain 24-hour period in winter was given by the following formula.     is time in hours and   corresponds to 6 A.M.

Accepted Answer

A) 13 < t < 24 
B) 0 < t < 11 
C) 0 < t < 24 
D) -13 < t < -24 
E) 0 < t < 13 
A) 13 < t < 24 
B) 0 < t < 11 
C) 0 < t < 24 
D) -13 < t < -24 
E) 0 < t < 13

Question 15

Express the statement as a formula that involves the variables q, x, y and a constant of proportionality k, and then determine the value of k from the condition : q is inversely proportional to the sum of x and y, if x = 1.9 and y = 3.7, then q = 0.7

Accepted Answer

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

Question 16

The period P of a simple pendulum - that is, the time required for one complete oscillation - is directly proportional to the square root of its length l. Express P in terms of l and a constant of proportionality k. If a pendulum 2.7 feet long has a period of 2.3 seconds, find the value of k.

Accepted Answer

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

Question 17

Does there exist a polynomial of degree 3 with real coefficients that has zeros 7 , - 7, and i ?

Accepted Answer

A) no 
B) yes 
A) no 
B) yes

Question 18

Salt water of concentration 0.5 pound of salt per gallon flows into a large tank that initially contains 220 gallons of pure water. If the flow rate of salt water into the tank is 4 gal/min, find a formula for the salt concentration   (in lb/gal) after t minutes.

Accepted Answer

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

Question 19

The period P of a simple pendulum - that is, the time required for one complete oscillation - is directly proportional to the square root of its length l. Express P in terms of l and a constant of proportionality k. If a pendulum 3.5 feet long has a period of 1.3 seconds, find the value of k.

Accepted Answer

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

Question 20

A herd of 575 deer is introduced onto a small island. At first the herd increases rapidly, but eventually food resources dwindle and the population declines. Suppose that the number of deer after   years is given by the following formula.   where   How many years does it take for the population to become extinct?

Accepted Answer

A) t = 4 
B) t = 6 
C) t = 10 
D) t = 8 
E) t = 5 
A) t = 4 
B) t = 6 
C) t = 10 
D) t = 8 
E) t = 5

Find all values of such that

Express the statement as a formula that involves the variables u, v and a constant of proportionality k, and determine the value of k from the condition : u is directly proportional to v and if v = 40, then u = 3

Applying the first theorem on bounds for real zeros of polynomials, determine the smallest and largest integers that are upper and lower bounds, respectively, for the real solutions of the equation. With the aid of a graphing utility, discuss the validity of the bounds.

Find the quotient and remainder if is divided by

Find all values of such that

A polynomial f ( x ) with real coefficients and leading coefficient 1 has the given zeros and degree. Express f ( x ) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R.

Find all solutions of the equation.

Find a polynomial with leading coefficient of , degree , and zeros: - .

Use the factor theorem to decide whether is a factor of the polynomial. ;

Use synthetic division to decide whether is a zero of . ;

A meteorologist determines that the temperature T for a certain 24-hour period in winter was given by the following formula. is time in hours and corresponds to 6 A.M.

Express the statement as a formula that involves the variables q, x, y and a constant of proportionality k, and then determine the value of k from the condition : q is inversely proportional to the sum of x and y, if x = 1.9 and y = 3.7, then q = 0.7

The period P of a simple pendulum - that is, the time required for one complete oscillation - is directly proportional to the square root of its length l. Express P in terms of l and a constant of proportionality k. If a pendulum 2.7 feet long has a period of 2.3 seconds, find the value of k.

Does there exist a polynomial of degree 3 with real coefficients that has zeros 7 , - 7, and i ?

Salt water of concentration 0.5 pound of salt per gallon flows into a large tank that initially contains 220 gallons of pure water. If the flow rate of salt water into the tank is 4 gal/min, find a formula for the salt concentration (in lb/gal) after t minutes.

The period P of a simple pendulum - that is, the time required for one complete oscillation - is directly proportional to the square root of its length l. Express P in terms of l and a constant of proportionality k. If a pendulum 3.5 feet long has a period of 1.3 seconds, find the value of k.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Inverse, Exponential, and Logarithmic Functions

The Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems of Equations and Inequalities

Sequences, Series and Probability

Topics From Analytic Geometry

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Exam 4: Polynomial and Rational Functions

Find all values of such that

Express the statement as a formula that involves the variables u, v and a constant of proportionality k, and determine the value of k from the condition : u is directly proportional to v and if v = 40, then u = 3

Applying the first theorem on bounds for real zeros of polynomials, determine the smallest and largest integers that are upper and lower bounds, respectively, for the real solutions of the equation. With the aid of a graphing utility, discuss the validity of the bounds.

Find the quotient and remainder if is divided by

Find all values of such that

A polynomial f ( x ) with real coefficients and leading coefficient 1 has the given zeros and degree. Express f ( x ) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R.

Find all solutions of the equation.

Find a polynomial with leading coefficient of , degree , and zeros: - .

Use the factor theorem to decide whether is a factor of the polynomial. ;

Use synthetic division to decide whether is a zero of . ;

A meteorologist determines that the temperature T for a certain 24-hour period in winter was given by the following formula. is time in hours and corresponds to 6 A.M.

Express the statement as a formula that involves the variables q, x, y and a constant of proportionality k, and then determine the value of k from the condition : q is inversely proportional to the sum of x and y, if x = 1.9 and y = 3.7, then q = 0.7

The period P of a simple pendulum - that is, the time required for one complete oscillation - is directly proportional to the square root of its length l. Express P in terms of l and a constant of proportionality k. If a pendulum 2.7 feet long has a period of 2.3 seconds, find the value of k.

Does there exist a polynomial of degree 3 with real coefficients that has zeros 7 , - 7, and i ?

Salt water of concentration 0.5 pound of salt per gallon flows into a large tank that initially contains 220 gallons of pure water. If the flow rate of salt water into the tank is 4 gal/min, find a formula for the salt concentration (in lb/gal) after t minutes.

The period P of a simple pendulum - that is, the time required for one complete oscillation - is directly proportional to the square root of its length l. Express P in terms of l and a constant of proportionality k. If a pendulum 3.5 feet long has a period of 1.3 seconds, find the value of k.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Inverse, Exponential, and Logarithmic Functions

The Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems of Equations and Inequalities

Sequences, Series and Probability

Topics From Analytic Geometry

Filters