Question 1

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

Accepted Answer

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

Question 2

Which of the following is the graph of the given equation? $f ( x ) = \frac { x - 2 } { x - 3 }$

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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)  $x ^ { 2 } + 3 x - 3 - \frac { 4 } { x ^ { 2 } + 3 x - 3 }$ 
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 5

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

Accepted Answer

A)  $a = - 3 , b = - 8$ 
B)  $a = 3 , b = - 8$ 
C)  $a = - 3 , b = 8$ 
D)  $a = 3 , b = 8$ 
E)  $a = - 5 , b = 11$ 
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 6

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?

Accepted Answer

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

Question 7

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

Accepted Answer

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. 
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

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

Accepted Answer

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)  $\pm 1 , \pm 2 , \pm 7 , \pm 14 , \pm \frac { 1 } { 11 } , \pm \frac { 2 } { 11 } , \pm \frac { 7 } { 11 } , \pm \frac { 14 } { 11 }$ 
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 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

Given the graph of $f ( x ) = \frac { 4 } { x ^ { 2 } }$ below, determine which of the following graphs represents $g ( x ) = \frac { 4 } { ( x + 3 ) ^ { 2 } } - 2$ .

Accepted Answer

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

Question 12

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 $\mathrm { V } : f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + 15 x + 9$

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

Use the zero $5 i$ of $f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } + 50 x + 75$ to find all the zeros of f.

Accepted Answer

A)  $- 5 i , \pm \frac { 3 } { 2 }$ 
B)  $- 5 i , - \frac { 3 } { 2 }$ 
C)  $\pm 5 i , - \frac { 3 } { 2 }$ 
D)  $5 i , - \frac { 3 } { 2 }$ 
E)  $5 i , \frac { 3 } { 2 }$ 
A)  $- 5 i , \pm \frac { 3 } { 2 }$ 
B)  $- 5 i , - \frac { 3 } { 2 }$ 
C)  $\pm 5 i , - \frac { 3 } { 2 }$ 
D)  $5 i , - \frac { 3 } { 2 }$ 
E)  $5 i , \frac { 3 } { 2 }$

Question 15

Determine the domain of $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 25 }$ .

Accepted Answer

A) all real numbers except $x = - 25$ 
B) all real numbers except $x = - 5 \text { and } x = 5$ 
C) all real numbers except $x = - 5 \text { and } x = 1$ 
D) all real numbers except $x = 5$ 
E) all real numbers 
A) all real numbers except $x = - 25$ 
B) all real numbers except $x = - 5 \text { and } x = 5$ 
C) all real numbers except $x = - 5 \text { and } x = 1$ 
D) all real numbers except $x = 5$ 
E) all real numbers

Question 16

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

Accepted Answer

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

Question 17

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.

Accepted Answer

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)  $8 ^ { \prime } \times 96 ^ { \prime }$ 
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 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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

Sketch the graph of the polynomial function below. P(x) =-10+36-54x+27 =(x-3)(x-3)(x-3)(x-1)

Which of the following is the graph of the given equation? $f ( x ) = \frac { x - 2 } { x - 3 }$

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

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

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

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

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

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

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

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

Use the zero $5 i$ of $f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } + 50 x + 75$ to find all the zeros of f.

Determine the domain of $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 25 }$ .

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

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.

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

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

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

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Series and Taylor Polynomials Web

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Exam 4: Polynomial and Rational Functions

Sketch the graph of the polynomial function below. P(x) =-10+36-54x+27 =(x-3)(x-3)(x-3)(x-1)

Which of the following is the graph of the given equation? f(x)=x−2x−3f ( x ) = \frac { x - 2 } { x - 3 }f(x)=x−3x−2​

Determine the vertex of the graph of the quadratic function f(x)=x2−8f ( x ) = x ^ { 2 } - 8f(x)=x2−8 .

Use long division to divide. (x4−2x2−5)÷(x2−3x−2)\left( x ^ { 4 } - 2 x ^ { 2 } - 5 \right) \div \left( x ^ { 2 } - 3 x - 2 \right)(x4−2x2−5)÷(x2−3x−2)

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

Describe the right-hand and the left-hand behavior of the graph of p(x)=6x5+10x3−27p ( x ) = 6 x ^ { 5 } + 10 x ^ { 3 } - 27p(x)=6x5+10x3−27 .

Use the Rational Zero Test to list all possible rational zeros of the polynomial function P(x)=11x3−x2+8x−14P ( x ) = 11 x ^ { 3 } - x ^ { 2 } + 8 x - 14P(x)=11x3−x2+8x−14 .

Use synthetic division and the Factor Theorem to determine whether or not 4x+34 x + 34x+3 is a factor of P(x)=−12x3−40x2+52x+80P ( x ) = - 12 x ^ { 3 } - 40 x ^ { 2 } + 52 x + 80P(x)=−12x3−40x2+52x+80 .

Use synthetic division and the Factor Theorem to determine whether or not x−4x - 4x−4 is a factor of P(x)=x3+2x2−x−2P ( x ) = x ^ { 3 } + 2 x ^ { 2 } - x - 2P(x)=x3+2x2−x−2 .

Given the graph of f(x)=4x2f ( x ) = \frac { 4 } { x ^ { 2 } }f(x)=x24​ below, determine which of the following graphs represents g(x)=4(x+3)2−2g ( x ) = \frac { 4 } { ( x + 3 ) ^ { 2 } } - 2g(x)=(x+3)24​−2 .

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

Use the zero 5i5 i5i of f(x)=2x3+3x2+50x+75f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } + 50 x + 75f(x)=2x3+3x2+50x+75 to find all the zeros of f.

Determine the domain of f(x)=x2x2+25f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 25 }f(x)=x2+25x2​ .

Determine the number of zeros of the polynomial function below. f(x)=9+x2−7x5+7x7f ( x ) = 9 + x ^ { 2 } - 7 x ^ { 5 } + 7 x ^ { 7 }f(x)=9+x2−7x5+7x7

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.

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

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

Sketch the graph of the polynomial function P(x)=−x3−x2+6xP ( x ) = - x ^ { 3 } - x ^ { 2 } + 6 xP(x)=−x3−x2+6x . Do not use a graphing calculator.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

Filters

Which of the following is the graph of the given equation? $f ( x ) = \frac { x - 2 } { x - 3 }$

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

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

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

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

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

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

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

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

Use the zero $5 i$ of $f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } + 50 x + 75$ to find all the zeros of f.

Determine the domain of $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 25 }$ .

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

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

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

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