Question 1

Use the factor theorem to decide whether or not the second polynomial is a factor of the first.
-$2 x ^ { 2 } - 14 x + 60 ; x - 3$

Accepted Answer

A) True 
 B)False

Question 2

Factor f(x) into linear factors given that k is a zero of f(x).
-$$f ( x ) = x ^ { 3 } - ( 4 + 2 i ) x ^ { 2 } + ( - 5 + 8 i ) x + 10 i ; k = 2 i$$

Accepted Answer

A)  $f ( x ) = ( x - i ) ( x + 1 ) ( x - 10 )$ 
B)  $f ( x ) = ( x - 2 i ) ( x - 1 ) ( x + 5 )$ 
C)  $f ( x ) = ( x + 2 i ) ( x - 1 ) ( x - 5 )$ 
D)  $f ( x ) = ( x - 2 i ) ( x + 1 ) ( x - 5 )$ 
A)  $f ( x ) = ( x - i ) ( x + 1 ) ( x - 10 )$ 
B)  $f ( x ) = ( x - 2 i ) ( x - 1 ) ( x + 5 )$ 
C)  $f ( x ) = ( x + 2 i ) ( x - 1 ) ( x - 5 )$ 
D)  $f ( x ) = ( x - 2 i ) ( x + 1 ) ( x - 5 )$

Question 3

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition.
-The polynomial f(x) $$f ( x ) = x ^ { 4 } - x ^ { 3 } + 2 x ^ { 2 } - 4 x - 10$$ has no real zero greater than 3.

Accepted Answer

A)  Yes, the boundedness theorem shows that the polynomial has no real zero greater than 3. 
B)  No, the boundedness theorem does not show that the polynomial has no real zero greater than 3. 
A)  Yes, the boundedness theorem shows that the polynomial has no real zero greater than 3. 
B)  No, the boundedness theorem does not show that the polynomial has no real zero greater than 3.

Question 4

Use the remainder theorem and synthetic division to find f(k).
-$k = 3 + i ; f ( x ) = x ^ { 3 } + 9$

Accepted Answer

A)   27+26 i 
B)   18+27 i 
C)   27+27 i 
D)   18+26 i 
A)   27+26 i 
B)   18+27 i 
C)   27+27 i 
D)   18+26 i

Question 5

Use synthetic division to perform the division.
-$$\frac { 2 x ^ { 3 } - 13 x ^ { 2 } + 23 x - 12 } { x - 4 }$$

Accepted Answer

A)  $- 2 x ^ { 2 } + 4 x + 3$ 
B)  $2 x - 5$ 
C)  $2 x ^ { 2 } - 5 x + 3$ 
D)  $\frac { 1 } { 2 } x ^ { 2 } - \frac { 13 } { 4 } x + \frac { 23 } { 4 }$ 
A)  $- 2 x ^ { 2 } + 4 x + 3$ 
B)  $2 x - 5$ 
C)  $2 x ^ { 2 } - 5 x + 3$ 
D)  $\frac { 1 } { 2 } x ^ { 2 } - \frac { 13 } { 4 } x + \frac { 23 } { 4 }$

Question 6

Sketch the graph of the rational function.
-$$f ( x ) = \frac { 5 x } { ( x - 5 ) ( x - 1 ) }$$

Accepted Answer

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

Question 7

Solve the problem. Round your answer to two decimal places.
-The weight of a liquid varies directly as its volume V. If the weight of the liquid in a cubical container 5 cm on a side is 375 g, find the weight of the liquid in a cubical container 3 cm on a side.

Accepted Answer

A)  12 g 
B)  9 g 
C)  81 g 
D)  27 g 
A)  12 g 
B)  9 g 
C)  81 g 
D)  27 g

Question 8

Find a polynomial of least degree with only real coefficients and having the given zeros.
-$3 , - 11$, and $4 + 4 i$

Accepted Answer

A)  $f ( x ) = x ^ { 4 } - 65 x ^ { 2 } + 520 x - 1056$ 
B)  $f ( x ) = x ^ { 4 } - 130 x ^ { 2 } + 520 x - 1056$ 
C)  $f ( x ) = x ^ { 4 } - 7 x ^ { 3 } + 12 x ^ { 2 } - 260 x + 1056$ 
D)  $f ( x ) = x ^ { 4 } - 7 x ^ { 3 } - 12 x ^ { 2 } + 260 x - 1056$ 
A)  $f ( x ) = x ^ { 4 } - 65 x ^ { 2 } + 520 x - 1056$ 
B)  $f ( x ) = x ^ { 4 } - 130 x ^ { 2 } + 520 x - 1056$ 
C)  $f ( x ) = x ^ { 4 } - 7 x ^ { 3 } + 12 x ^ { 2 } - 260 x + 1056$ 
D)  $f ( x ) = x ^ { 4 } - 7 x ^ { 3 } - 12 x ^ { 2 } + 260 x - 1056$

Question 9

Use the equation and the corresponding graph for the quadratic function to find what is requested.
-$f(x)=(x-3)^{2}-4$

Find the $x$-intercepts.

Accepted Answer

A)  $( 3 , - 4 )$ 
B)  $( 1,0 )$ and $( 5,0 )$ 
C)  $( 2,0 )$ and $( 6,0 )$ 
D)  $( 0,1 )$ and $( 0,5 )$ 
A)  $( 3 , - 4 )$ 
B)  $( 1,0 )$ and $( 5,0 )$ 
C)  $( 2,0 )$ and $( 6,0 )$ 
D)  $( 0,1 )$ and $( 0,5 )$

Question 10

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function.
-$$f ( x ) = - 2 x ^ { 4 } - 4 x ^ { 3 } - 4 x ^ { 2 } - 3 x + 8$$

Accepted Answer

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

Question 11

x and y are two positive numbers and y is three greater than x. The product of the numbers is 130. Find the smaller number, x.

Accepted Answer

A)  5 
B)  10 
C)  7 
D)  13 
A)  5 
B)  10 
C)  7 
D)  13

Question 12

$$\text { How can the graph of } f ( x ) = - \frac { 1 } { x ^ { 2 } } - 6 \text { be obtained from the graph of } y = \frac { 1 } { x ^ { 2 } } ?$$

Accepted Answer

A)  By making a horizontal shift of 1 unit to the right and a vertical shift of 6 units up 
B)  By reflecting across the x-axis and making a vertical shift of 6 units down 
C)  By making a horizontal shift of 6 units to the left and a vertical shift of 1 unit down 
D)  By reflecting across the y-axis and making a horizontal shift of 6 units to the left 
A)  By making a horizontal shift of 1 unit to the right and a vertical shift of 6 units up 
B)  By reflecting across the x-axis and making a vertical shift of 6 units down 
C)  By making a horizontal shift of 6 units to the left and a vertical shift of 1 unit down 
D)  By reflecting across the y-axis and making a horizontal shift of 6 units to the left

Question 13

Determine whether the statement is true or false.
-$f(x)=\frac{2}{x+3}$

Accepted Answer

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

Question 14

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function.
-$$f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 8 x + 2$$

Accepted Answer

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

Question 15

Give the equations of any horizontal asymptotes.
-$$h ( x ) = \frac { 27 x ^ { 2 } } { 9 x ^ { 2 } - 8 }$$

Accepted Answer

A)  $y = \sqrt { 8 }$ 
B)  $y = 3$ 
C)  $y = 8$ 
D)  none 
A)  $y = \sqrt { 8 }$ 
B)  $y = 3$ 
C)  $y = 8$ 
D)  none

Question 16

$$\text { If } \mathrm { f } \text { varies jointly as } \mathrm { q } ^ { 2 } \text { and } h \text {, and } \mathrm { f } = 36 \text { when } \mathrm { q } = 2 \text { and } h = 3 \text {, find } \mathrm { f } \text { when } \mathrm { q } = 4 \text { and } \mathrm { h } = 6 \text {. }$$

Accepted Answer

A)  288 
B)  18 
C)  48 
D)  72 
A)  288 
B)  18 
C)  48 
D)  72

Question 17

The graph of $f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - x - 4$ is shown below. Use the graph to factor $f ( x )$.

Accepted Answer

A)  $f ( x ) = ( x + 4 ) ( x + 1 ) ( x - 1 )$ 
B)  $f ( x ) = ( x - 4 ) ( x + 1 ) ( x + 4 )$ 
C)  $f ( x ) = ( x - 4 ) ( x + 1 ) ( x - 1 )$ 
D)  $f ( x ) = - ( x + 4 ) ( x + 1 ) ( x - 1 )$ 
A)  $f ( x ) = ( x + 4 ) ( x + 1 ) ( x - 1 )$ 
B)  $f ( x ) = ( x - 4 ) ( x + 1 ) ( x + 4 )$ 
C)  $f ( x ) = ( x - 4 ) ( x + 1 ) ( x - 1 )$ 
D)  $f ( x ) = - ( x + 4 ) ( x + 1 ) ( x - 1 )$

Question 18

Find the domain and range of the function.
$f ( x ) = x ^ { 2 } - 18 x + 84$

Accepted Answer

A)  $\text { Domain: } [ 3 , \infty ) \text {; Range: } ( - \infty , 9 ]$ 
B)  $\text { Domain: } ( - \infty , \infty ) \text {; Range: } [ 3 , \infty )$ 
C)  $\text { Domain: } [ 3 , \infty ) \text {; Range: } ( - \infty , 0 ]$ 
D)  $\text { Domain: } ( - \infty , \infty ) \text {; Range: } [ 9 , \infty )$ 
A)  $\text { Domain: } [ 3 , \infty ) \text {; Range: } ( - \infty , 9 ]$ 
B)  $\text { Domain: } ( - \infty , \infty ) \text {; Range: } [ 3 , \infty )$ 
C)  $\text { Domain: } [ 3 , \infty ) \text {; Range: } ( - \infty , 0 ]$ 
D)  $\text { Domain: } ( - \infty , \infty ) \text {; Range: } [ 9 , \infty )$

Question 19

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition.
-The polynomial f(x) $f ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 x$ has no real zero less than -8.

Accepted Answer

A)  Yes, the boundedness theorem shows that the polynomial has no real zero less than -8. 
B)  No, the boundedness theorem does not show that the polynomial has no real zero less than -8. 
A)  Yes, the boundedness theorem shows that the polynomial has no real zero less than -8. 
B)  No, the boundedness theorem does not show that the polynomial has no real zero less than -8.

Question 20

Why does a function defined by a polynomial of degree five with real coefficients have either 1, 3, or 5 real zeros
counting multiplicities?

Accepted Answer

By the Fundamental Theorem of Algebra an

Use the factor theorem to decide whether or not the second polynomial is a factor of the first. - $2 x ^ { 2 } - 14 x + 60 ; x - 3$

Factor f(x) into linear factors given that k is a zero of f(x). - $f ( x ) = x ^ { 3 } - ( 4 + 2 i ) x ^ { 2 } + ( - 5 + 8 i ) x + 10 i ; k = 2 i$

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition. -The polynomial f(x) $f ( x ) = x ^ { 4 } - x ^ { 3 } + 2 x ^ { 2 } - 4 x - 10$ has no real zero greater than 3.

Use the remainder theorem and synthetic division to find f(k). - $k = 3 + i ; f ( x ) = x ^ { 3 } + 9$

Use synthetic division to perform the division. - $\frac { 2 x ^ { 3 } - 13 x ^ { 2 } + 23 x - 12 } { x - 4 }$

Sketch the graph of the rational function. - $f ( x ) = \frac { 5 x } { ( x - 5 ) ( x - 1 ) }$ $Sketch the graph of the rational function. - f ( x ) = \frac { 5 x } { ( x - 5 ) ( x - 1 ) }$

Solve the problem. Round your answer to two decimal places. -The weight of a liquid varies directly as its volume V. If the weight of the liquid in a cubical container 5 cm on a side is 375 g, find the weight of the liquid in a cubical container 3 cm on a side.

Find a polynomial of least degree with only real coefficients and having the given zeros. - $3 , - 11$ , and $4 + 4 i$

Use the equation and the corresponding graph for the quadratic function to find what is requested. - $f(x)=(x-3)^{2}-4$ $Use the equation and the corresponding graph for the quadratic function to find what is requested. - f(x)=(x-3)^{2}-4 Find the x -intercepts.$ Find the $x$ -intercepts.

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = - 2 x ^ { 4 } - 4 x ^ { 3 } - 4 x ^ { 2 } - 3 x + 8$

x and y are two positive numbers and y is three greater than x. The product of the numbers is 130. Find the smaller number, x.

$\text { How can the graph of } f ( x ) = - \frac { 1 } { x ^ { 2 } } - 6 \text { be obtained from the graph of } y = \frac { 1 } { x ^ { 2 } } ?$

Determine whether the statement is true or false. - $f(x)=\frac{2}{x+3}$ $Determine whether the statement is true or false. - f(x)=\frac{2}{x+3}$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 8 x + 2$

Give the equations of any horizontal asymptotes. - $h ( x ) = \frac { 27 x ^ { 2 } } { 9 x ^ { 2 } - 8 }$

$\text { If } \mathrm { f } \text { varies jointly as } \mathrm { q } ^ { 2 } \text { and } h \text {, and } \mathrm { f } = 36 \text { when } \mathrm { q } = 2 \text { and } h = 3 \text {, find } \mathrm { f } \text { when } \mathrm { q } = 4 \text { and } \mathrm { h } = 6 \text {. }$

The graph of $f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - x - 4$ is shown below. Use the graph to factor $f ( x )$ . $The graph of f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - x - 4 is shown below. Use the graph to factor f ( x ) .$

Find the domain and range of the function. $f ( x ) = x ^ { 2 } - 18 x + 84$

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition. -The polynomial f(x) $f ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 x$ has no real zero less than -8.

Why does a function defined by a polynomial of degree five with real coefficients have either 1, 3, or 5 real zeros counting multiplicities?

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

Arithmetic Sequence: Common Difference and First n Terms

Filters

Exam 4: Polynomials and Rational Functions

Use the factor theorem to decide whether or not the second polynomial is a factor of the first. - 2x2−14x+60;x−32 x ^ { 2 } - 14 x + 60 ; x - 32x2−14x+60;x−3

Factor f(x) into linear factors given that k is a zero of f(x). - f(x)=x3−(4+2i)x2+(−5+8i)x+10i;k=2if ( x ) = x ^ { 3 } - ( 4 + 2 i ) x ^ { 2 } + ( - 5 + 8 i ) x + 10 i ; k = 2 if(x)=x3−(4+2i)x2+(−5+8i)x+10i;k=2i

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition. -The polynomial f(x) f(x)=x4−x3+2x2−4x−10f ( x ) = x ^ { 4 } - x ^ { 3 } + 2 x ^ { 2 } - 4 x - 10f(x)=x4−x3+2x2−4x−10 has no real zero greater than 3.

Use the remainder theorem and synthetic division to find f(k). - k=3+i;f(x)=x3+9k = 3 + i ; f ( x ) = x ^ { 3 } + 9k=3+i;f(x)=x3+9

Use synthetic division to perform the division. - 2x3−13x2+23x−12x−4\frac { 2 x ^ { 3 } - 13 x ^ { 2 } + 23 x - 12 } { x - 4 }x−42x3−13x2+23x−12​

Sketch the graph of the rational function. - f(x)=5x(x−5)(x−1)f ( x ) = \frac { 5 x } { ( x - 5 ) ( x - 1 ) }f(x)=(x−5)(x−1)5x​

Solve the problem. Round your answer to two decimal places. -The weight of a liquid varies directly as its volume V. If the weight of the liquid in a cubical container 5 cm on a side is 375 g, find the weight of the liquid in a cubical container 3 cm on a side.

Find a polynomial of least degree with only real coefficients and having the given zeros. - 3,−113 , - 113,−11 , and 4+4i4 + 4 i4+4i

Use the equation and the corresponding graph for the quadratic function to find what is requested. - f(x)=(x−3)2−4f(x)=(x-3)^{2}-4f(x)=(x−3)2−4 Find the xxx -intercepts.

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - f(x)=−2x4−4x3−4x2−3x+8f ( x ) = - 2 x ^ { 4 } - 4 x ^ { 3 } - 4 x ^ { 2 } - 3 x + 8f(x)=−2x4−4x3−4x2−3x+8

x and y are two positive numbers and y is three greater than x. The product of the numbers is 130. Find the smaller number, x.

How can the graph of f(x)=−1x2−6 be obtained from the graph of y=1x2?\text { How can the graph of } f ( x ) = - \frac { 1 } { x ^ { 2 } } - 6 \text { be obtained from the graph of } y = \frac { 1 } { x ^ { 2 } } ? How can the graph of f(x)=−x21​−6 be obtained from the graph of y=x21​?

Determine whether the statement is true or false. - f(x)=2x+3f(x)=\frac{2}{x+3}f(x)=x+32​

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - f(x)=2x3−8x2+8x+2f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 8 x + 2f(x)=2x3−8x2+8x+2

Give the equations of any horizontal asymptotes. - h(x)=27x29x2−8h ( x ) = \frac { 27 x ^ { 2 } } { 9 x ^ { 2 } - 8 }h(x)=9x2−827x2​

The graph of f(x)=x3+4x2−x−4f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - x - 4f(x)=x3+4x2−x−4 is shown below. Use the graph to factor f(x)f ( x )f(x) .

Find the domain and range of the function. f(x)=x2−18x+84f ( x ) = x ^ { 2 } - 18 x + 84f(x)=x2−18x+84

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition. -The polynomial f(x) f(x)=x5+10x4−25x3−10x2+24xf ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 xf(x)=x5+10x4−25x3−10x2+24x has no real zero less than -8.

Why does a function defined by a polynomial of degree five with real coefficients have either 1, 3, or 5 real zeros counting multiplicities?

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

Arithmetic Sequence: Common Difference and First n Terms

Filters

Use the factor theorem to decide whether or not the second polynomial is a factor of the first. - $2 x ^ { 2 } - 14 x + 60 ; x - 3$

Factor f(x) into linear factors given that k is a zero of f(x). - $f ( x ) = x ^ { 3 } - ( 4 + 2 i ) x ^ { 2 } + ( - 5 + 8 i ) x + 10 i ; k = 2 i$

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition. -The polynomial f(x) $f ( x ) = x ^ { 4 } - x ^ { 3 } + 2 x ^ { 2 } - 4 x - 10$ has no real zero greater than 3.

Use the remainder theorem and synthetic division to find f(k). - $k = 3 + i ; f ( x ) = x ^ { 3 } + 9$

Use synthetic division to perform the division. - $\frac { 2 x ^ { 3 } - 13 x ^ { 2 } + 23 x - 12 } { x - 4 }$

Sketch the graph of the rational function. - $f ( x ) = \frac { 5 x } { ( x - 5 ) ( x - 1 ) }$ $Sketch the graph of the rational function. - f ( x ) = \frac { 5 x } { ( x - 5 ) ( x - 1 ) }$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $3 , - 11$ , and $4 + 4 i$

Use the equation and the corresponding graph for the quadratic function to find what is requested. - $f(x)=(x-3)^{2}-4$ $Use the equation and the corresponding graph for the quadratic function to find what is requested. - f(x)=(x-3)^{2}-4 Find the x -intercepts.$ Find the $x$ -intercepts.

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = - 2 x ^ { 4 } - 4 x ^ { 3 } - 4 x ^ { 2 } - 3 x + 8$

$\text { How can the graph of } f ( x ) = - \frac { 1 } { x ^ { 2 } } - 6 \text { be obtained from the graph of } y = \frac { 1 } { x ^ { 2 } } ?$

Determine whether the statement is true or false. - $f(x)=\frac{2}{x+3}$ $Determine whether the statement is true or false. - f(x)=\frac{2}{x+3}$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 8 x + 2$

Give the equations of any horizontal asymptotes. - $h ( x ) = \frac { 27 x ^ { 2 } } { 9 x ^ { 2 } - 8 }$

The graph of $f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - x - 4$ is shown below. Use the graph to factor $f ( x )$ . $The graph of f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - x - 4 is shown below. Use the graph to factor f ( x ) .$

Find the domain and range of the function. $f ( x ) = x ^ { 2 } - 18 x + 84$

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition. -The polynomial f(x) $f ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 x$ has no real zero less than -8.