Question 1

Find an equation for the rational function graph.
-

Accepted Answer

A)  $f(x)=\frac{x(x+2)}{(x+2)^{2}}$ 
B)  $f(x)=\frac{x(x-2)}{(x-2)^{2}}$ 
C)  $f ( x ) = \frac { ( x + 2 ) ( x - 2 ) } { x - 2 }$ 
D)  $f ( x ) = \frac { x ( x - 2 ) } { x - 2 }$ 
A)  $f(x)=\frac{x(x+2)}{(x+2)^{2}}$ 
B)  $f(x)=\frac{x(x-2)}{(x-2)^{2}}$ 
C)  $f ( x ) = \frac { ( x + 2 ) ( x - 2 ) } { x - 2 }$ 
D)  $f ( x ) = \frac { x ( x - 2 ) } { x - 2 }$

Question 2

Find the equation that the given graph represents.
-

Accepted Answer

A)  $P ( x ) = - x ^ { 5 } - 20 x ^ { 4 } - 100 x ^ { 2 } + 100 x$ 
B)  $P ( x ) = - x ^ { 6 } + 10 x ^ { 4 } - 100 x ^ { 2 } - 100$ 
C)  $P ( x ) = - x ^ { 6 } + 20 x ^ { 4 } - 100 x ^ { 2 } + 100$ 
D)  $P ( x ) = x ^ { 5 } - 10 x ^ { 4 } - 100 x ^ { 2 } + 100$ 
A)  $P ( x ) = - x ^ { 5 } - 20 x ^ { 4 } - 100 x ^ { 2 } + 100 x$ 
B)  $P ( x ) = - x ^ { 6 } + 10 x ^ { 4 } - 100 x ^ { 2 } - 100$ 
C)  $P ( x ) = - x ^ { 6 } + 20 x ^ { 4 } - 100 x ^ { 2 } + 100$ 
D)  $P ( x ) = x ^ { 5 } - 10 x ^ { 4 } - 100 x ^ { 2 } + 100$

Question 3

Use the boundedness theorem to determine whether the statement is true or false.
-The polynomial $f ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 x$ has no real zero greater than 3 .

Accepted Answer

A) True 
 B)False

Question 4

Solve the problem. Round to the nearest tenth unless indicated otherwise.
-The volume of a gas varies inversely as the pressure and directly as the temperature (in degrees Kelvin). If a certain gas occupies a volume of $2.3$ liters at a temperature of $320 \mathrm {~K}$ and a pressure of 18 newtons per square centimeter, find the volume when the temperature is $288 \mathrm {~K}$ and the pressure is 9 newtons per square centimeter.

Accepted Answer

A)  $4.1 \mathrm {~L}$ 
B)  $46.0 \mathrm {~L}$ 
C)  $40.9 \mathrm {~L}$ 
D)  $1.3 \mathrm {~L}$ 
A)  $4.1 \mathrm {~L}$ 
B)  $46.0 \mathrm {~L}$ 
C)  $40.9 \mathrm {~L}$ 
D)  $1.3 \mathrm {~L}$

Question 5

Use the boundedness theorem to determine whether the statement is true or false.
-The polynomial $f ( x ) = x ^ { 4 } - x ^ { 3 } + 2 x ^ { 2 } - 4 x - 10$ has no real zero less than $- 1$.

Accepted Answer

A) True 
 B)False

Question 6

Solve the problem.
-A can has a surface area of 970 square inches. Its height is $5.34$ inches. What is the radius of the circular top? Round to the nearest hundredth.

Accepted Answer

A)  $10.04 \mathrm { in }$. 
B)  $13.03$ in. 
C)  $28.91 \mathrm { in }$. 
D)  $4.93 \mathrm { in }$. 
A)  $10.04 \mathrm { in }$. 
B)  $13.03$ in. 
C)  $28.91 \mathrm { in }$. 
D)  $4.93 \mathrm { in }$.

Question 7

Give the equations of any vertical asymptotes.
-$$g ( x ) = \frac { x - 5 } { ( x - 4 ) ( x + 7 ) }$$

Accepted Answer

A)  $x = - 4 , x = 7$ 
B)  $x = 4 , x = - 7$ 
C)  $x = 5$ 
D)  $x = - 5$ 
A)  $x = - 4 , x = 7$ 
B)  $x = 4 , x = - 7$ 
C)  $x = 5$ 
D)  $x = - 5$

Question 8

Find a polynomial of least degree with only real coefficients and having the given zeros.
-$7 , - 11$, and $2 + 4 \mathrm { i }$

Accepted Answer

A)  $f ( x ) = x ^ { 4 } - 9 x ^ { 3 } - 28 x ^ { 2 } + 194 x - 1540$ 
B)  $f ( x ) = x ^ { 4 } - 73 x ^ { 2 } + 388 x - 1540$ 
C)  $f ( x ) = x ^ { 4 } - 97 x ^ { 2 } + 388 x - 1540$ 
D)  $f ( x ) = x ^ { 4 } - 9 x ^ { 3 } + 28 x ^ { 2 } - 194 x + 1540$ 
A)  $f ( x ) = x ^ { 4 } - 9 x ^ { 3 } - 28 x ^ { 2 } + 194 x - 1540$ 
B)  $f ( x ) = x ^ { 4 } - 73 x ^ { 2 } + 388 x - 1540$ 
C)  $f ( x ) = x ^ { 4 } - 97 x ^ { 2 } + 388 x - 1540$ 
D)  $f ( x ) = x ^ { 4 } - 9 x ^ { 3 } + 28 x ^ { 2 } - 194 x + 1540$

Question 9

Solve.
-The student population of a small school has been increasing as shown in the following table.

Determine the linear, quadratic, or cubic function that best fits the data.

Accepted Answer

A)  $f ( x ) = x ^ { 3 } - 0.1 x ^ { 2 } - 10 x + 440$ 
B)  $f ( x ) = 0.002 x ^ { 3 } - 0.44 x ^ { 2 } + 28.19 x - 427.30$ 
C)  $f ( x ) = 5 x - 93.33$ 
D)  $f ( x ) = 0.1 x ^ { 2 } - 10 x + 440$ 
A)  $f ( x ) = x ^ { 3 } - 0.1 x ^ { 2 } - 10 x + 440$ 
B)  $f ( x ) = 0.002 x ^ { 3 } - 0.44 x ^ { 2 } + 28.19 x - 427.30$ 
C)  $f ( x ) = 5 x - 93.33$ 
D)  $f ( x ) = 0.1 x ^ { 2 } - 10 x + 440$

Question 10

Use the remainder theorem and synthetic division to find f(k).
-$$\mathrm { k } = - 3 ; \mathrm { f } ( \mathrm { x } ) = 3 \mathrm { x } ^ { 3 } - 7 \mathrm { x } ^ { 2 } - 5 \mathrm { x } + 5$$

Accepted Answer

A)  48 
B)  $- 124$ 
C)  $- 38$ 
D)  $- 154$ 
A)  48 
B)  $- 124$ 
C)  $- 38$ 
D)  $- 154$

Question 11

Answer the question
-How can the graph of $f ( x ) = \frac { 6 } { ( x + 4 ) ^ { 2 } }$ by obtained from the graph of $y = \frac { 1 } { x ^ { 2 } }$ ?

Accepted Answer

A)  By making a horizontal shift of 4 units to the left and a vertical shift of 6 units down 
B)  By making a horizontal shift of 4 units to the right and stretching vertically by a factor of 6 
C)  By making a horizontal shift of 6 units to the right and a vertical shift of 4 units up 
D)  By making a horizontal shift of 4 units to the left and stretching vertically by a factor of 6 
A)  By making a horizontal shift of 4 units to the left and a vertical shift of 6 units down 
B)  By making a horizontal shift of 4 units to the right and stretching vertically by a factor of 6 
C)  By making a horizontal shift of 6 units to the right and a vertical shift of 4 units up 
D)  By making a horizontal shift of 4 units to the left and stretching vertically by a factor of 6

Question 12

Find the equation that the given graph represents.
-

Accepted Answer

A)  $P ( x ) = x ^ { 3 } + 9 x$ 
B)  $P ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 20 x$ 
C)  $P ( x ) = - x ^ { 3 } + 4 x ^ { 2 } - 9$ 
D)  $P ( x ) = - x ^ { 3 } + 20 x$ 
A)  $P ( x ) = x ^ { 3 } + 9 x$ 
B)  $P ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 20 x$ 
C)  $P ( x ) = - x ^ { 3 } + 4 x ^ { 2 } - 9$ 
D)  $P ( x ) = - x ^ { 3 } + 20 x$

Question 13

Use a graphing calculator to approximate the real zeros. Give each zero as a decimal to the nearest tenth.
-$$f ( x ) = 4 x ^ { 3 } + x ^ { 2 } - 20 x - 5$$

Accepted Answer

A)  $- 0.3,2.2 , - 2.2$ 
B)  $0.3,2.2 , - 2.2$ 
C)  $- 4,2.3 , - 2.3$ 
D)  $4,5 , - 5$ 
A)  $- 0.3,2.2 , - 2.2$ 
B)  $0.3,2.2 , - 2.2$ 
C)  $- 4,2.3 , - 2.3$ 
D)  $4,5 , - 5$

Question 14

Solve the problem.
-The table shows the population of a city over the past five years.

We used these data to develop the quadratic function $f ( x ) = 0.037214 x ^ { 2 } + 0.364 x + 65$, which models the population of the city $y$ in millions in the year $x$, where $x = 0$ represents 2007. Use the model to find the estimated population in $2010 .$

Accepted Answer

A)  $71,628,000$ 
B)  $66,427,000$ 
C)  $1,490,000$ 
D)  $1,490,000,000,000$ 
A)  $71,628,000$ 
B)  $66,427,000$ 
C)  $1,490,000$ 
D)  $1,490,000,000,000$

Question 15

Solve the problem.
-$f(x)=-(x-4)^{4}$

Accepted Answer

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

Question 16

Provide an appropriate response.
-A quadratic equation $f ( x ) = 0$ has a solution $x = 3$. Its graph has vertex $( - 1 , - 16 )$. What is the other solution of the equation?

Accepted Answer

A)  7 
B)  $- 5$ 
C)  $- 4$ 
D)  $- 3$ 
A)  7 
B)  $- 5$ 
C)  $- 4$ 
D)  $- 3$

Question 17

Solve.
-The concentration of a certain gas molecule in the atmosphere serves as an indicator of industrial air pollution. The data in the following table show the relationship of the estimated concentration of the molecule in the atmosphere, in parts per billion ( $\mathrm { ppb }$ ), to the year.

Determine the linear, quadratic, or cubic function that best fits the data.

Accepted Answer

A)  $f ( x ) = 0.002 x + 2.90$ 
B)  $f ( x ) = 0.00001 x ^ { 3 } - 0.003 x ^ { 2 } + 0.14 x + 2.16$ 
C)  $f ( x ) = - 0.003 x ^ { 2 } + 0.12 x + 2.19$ 
D)  $f ( x ) = - 0.002 x ^ { 2 } + 0.12 x + 2.19$ 
A)  $f ( x ) = 0.002 x + 2.90$ 
B)  $f ( x ) = 0.00001 x ^ { 3 } - 0.003 x ^ { 2 } + 0.14 x + 2.16$ 
C)  $f ( x ) = - 0.003 x ^ { 2 } + 0.12 x + 2.19$ 
D)  $f ( x ) = - 0.002 x ^ { 2 } + 0.12 x + 2.19$

Question 18

Sketch the graph of the rational function.
-$$f ( x ) = \frac { ( x - 1 ) ( x - 3 ) } { ( x + 5 ) ^ { 2 } }$$

Accepted Answer

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

Question 19

Give all possible rational zeros for the following polynomial.
-$$P ( x ) = 3 x ^ { 3 } + 3 x ^ { 2 } + 2 x - 6$$

Accepted Answer

A)  $\pm 1 , \pm \frac { 1 } { 2 } , \pm 3 , \pm \frac { 3 } { 2 } , \pm \frac { 1 } { 3 } , \pm \frac { 1 } { 6 }$ 
B)  $\pm 1 , \pm \frac { 1 } { 3 } , \pm 2 , \pm 3 , \pm 6$ 
C)  $\pm 1 , \pm 2 , \pm 3 , \pm 6$ 
D)  $\pm 1 , \pm \frac { 1 } { 3 } , \pm 2 , \pm \frac { 2 } { 3 } , \pm 3 , \pm 6$ 
A)  $\pm 1 , \pm \frac { 1 } { 2 } , \pm 3 , \pm \frac { 3 } { 2 } , \pm \frac { 1 } { 3 } , \pm \frac { 1 } { 6 }$ 
B)  $\pm 1 , \pm \frac { 1 } { 3 } , \pm 2 , \pm 3 , \pm 6$ 
C)  $\pm 1 , \pm 2 , \pm 3 , \pm 6$ 
D)  $\pm 1 , \pm \frac { 1 } { 3 } , \pm 2 , \pm \frac { 2 } { 3 } , \pm 3 , \pm 6$

Question 20

Use the factor theorem to decide whether or not the second polynomial is a factor of the first.
-$$x ^ { 3 } - 5 x ^ { 2 } + 11 x - 15 ; x - 3$$

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Find an equation for the rational function graph. -

Find the equation that the given graph represents. -

Use the boundedness theorem to determine whether the statement is true or false. -The polynomial $f ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 x$ has no real zero greater than 3 .

Use the boundedness theorem to determine whether the statement is true or false. -The polynomial $f ( x ) = x ^ { 4 } - x ^ { 3 } + 2 x ^ { 2 } - 4 x - 10$ has no real zero less than $- 1$ .

Solve the problem. -A can has a surface area of 970 square inches. Its height is $5.34$ inches. What is the radius of the circular top? Round to the nearest hundredth.

Give the equations of any vertical asymptotes. - $g ( x ) = \frac { x - 5 } { ( x - 4 ) ( x + 7 ) }$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $7 , - 11$ , and $2 + 4 \mathrm { i }$

Solve. -The student population of a small school has been increasing as shown in the following table. Determine the linear, quadratic, or cubic function that best fits the data.

Use the remainder theorem and synthetic division to find f(k). - $\mathrm { k } = - 3 ; \mathrm { f } ( \mathrm { x } ) = 3 \mathrm { x } ^ { 3 } - 7 \mathrm { x } ^ { 2 } - 5 \mathrm { x } + 5$

Answer the question -How can the graph of $f ( x ) = \frac { 6 } { ( x + 4 ) ^ { 2 } }$ by obtained from the graph of $y = \frac { 1 } { x ^ { 2 } }$ ?

Find the equation that the given graph represents. -

Use a graphing calculator to approximate the real zeros. Give each zero as a decimal to the nearest tenth. - $f ( x ) = 4 x ^ { 3 } + x ^ { 2 } - 20 x - 5$

Solve the problem. - $f(x)=-(x-4)^{4}$ $Solve the problem. - f(x)=-(x-4)^{4}$

Provide an appropriate response. -A quadratic equation $f ( x ) = 0$ has a solution $x = 3$ . Its graph has vertex $( - 1 , - 16 )$ . What is the other solution of the equation?

Sketch the graph of the rational function. - $f ( x ) = \frac { ( x - 1 ) ( x - 3 ) } { ( x + 5 ) ^ { 2 } }$ $Sketch the graph of the rational function. - f ( x ) = \frac { ( x - 1 ) ( x - 3 ) } { ( x + 5 ) ^ { 2 } }$

Give all possible rational zeros for the following polynomial. - $P ( x ) = 3 x ^ { 3 } + 3 x ^ { 2 } + 2 x - 6$

Use the factor theorem to decide whether or not the second polynomial is a factor of the first. - $x ^ { 3 } - 5 x ^ { 2 } + 11 x - 15 ; x - 3$

Equations and Inequalities

Graphs and Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Review of Basic Concepts

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Exam 3: Polynomial and Rational Functions

Find an equation for the rational function graph. -

Find the equation that the given graph represents. -

Use the boundedness theorem to determine whether the statement is true or false. -The polynomial 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 greater than 3 .

Use the boundedness theorem to determine whether the statement is true or false. -The polynomial 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 less than −1- 1−1 .

Solve the problem. -A can has a surface area of 970 square inches. Its height is 5.345.345.34 inches. What is the radius of the circular top? Round to the nearest hundredth.

Give the equations of any vertical asymptotes. - g(x)=x−5(x−4)(x+7)g ( x ) = \frac { x - 5 } { ( x - 4 ) ( x + 7 ) }g(x)=(x−4)(x+7)x−5​

Find a polynomial of least degree with only real coefficients and having the given zeros. - 7,−117 , - 117,−11 , and 2+4i2 + 4 \mathrm { i }2+4i

Solve. -The student population of a small school has been increasing as shown in the following table. Determine the linear, quadratic, or cubic function that best fits the data.

Use the remainder theorem and synthetic division to find f(k). - k=−3;f(x)=3x3−7x2−5x+5\mathrm { k } = - 3 ; \mathrm { f } ( \mathrm { x } ) = 3 \mathrm { x } ^ { 3 } - 7 \mathrm { x } ^ { 2 } - 5 \mathrm { x } + 5k=−3;f(x)=3x3−7x2−5x+5

Answer the question -How can the graph of f(x)=6(x+4)2f ( x ) = \frac { 6 } { ( x + 4 ) ^ { 2 } }f(x)=(x+4)26​ by obtained from the graph of y=1x2y = \frac { 1 } { x ^ { 2 } }y=x21​ ?

Find the equation that the given graph represents. -

Use a graphing calculator to approximate the real zeros. Give each zero as a decimal to the nearest tenth. - f(x)=4x3+x2−20x−5f ( x ) = 4 x ^ { 3 } + x ^ { 2 } - 20 x - 5f(x)=4x3+x2−20x−5

Solve the problem. - f(x)=−(x−4)4f(x)=-(x-4)^{4}f(x)=−(x−4)4

Provide an appropriate response. -A quadratic equation f(x)=0f ( x ) = 0f(x)=0 has a solution x=3x = 3x=3 . Its graph has vertex (−1,−16)( - 1 , - 16 )(−1,−16) . What is the other solution of the equation?

Sketch the graph of the rational function. - f(x)=(x−1)(x−3)(x+5)2f ( x ) = \frac { ( x - 1 ) ( x - 3 ) } { ( x + 5 ) ^ { 2 } }f(x)=(x+5)2(x−1)(x−3)​

Give all possible rational zeros for the following polynomial. - P(x)=3x3+3x2+2x−6P ( x ) = 3 x ^ { 3 } + 3 x ^ { 2 } + 2 x - 6P(x)=3x3+3x2+2x−6

Use the factor theorem to decide whether or not the second polynomial is a factor of the first. - x3−5x2+11x−15;x−3x ^ { 3 } - 5 x ^ { 2 } + 11 x - 15 ; x - 3x3−5x2+11x−15;x−3

Equations and Inequalities

Graphs and Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Review of Basic Concepts

Filters

Use the boundedness theorem to determine whether the statement is true or false. -The polynomial $f ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 x$ has no real zero greater than 3 .

Use the boundedness theorem to determine whether the statement is true or false. -The polynomial $f ( x ) = x ^ { 4 } - x ^ { 3 } + 2 x ^ { 2 } - 4 x - 10$ has no real zero less than $- 1$ .

Solve the problem. -A can has a surface area of 970 square inches. Its height is $5.34$ inches. What is the radius of the circular top? Round to the nearest hundredth.

Give the equations of any vertical asymptotes. - $g ( x ) = \frac { x - 5 } { ( x - 4 ) ( x + 7 ) }$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $7 , - 11$ , and $2 + 4 \mathrm { i }$

Use the remainder theorem and synthetic division to find f(k). - $\mathrm { k } = - 3 ; \mathrm { f } ( \mathrm { x } ) = 3 \mathrm { x } ^ { 3 } - 7 \mathrm { x } ^ { 2 } - 5 \mathrm { x } + 5$

Answer the question -How can the graph of $f ( x ) = \frac { 6 } { ( x + 4 ) ^ { 2 } }$ by obtained from the graph of $y = \frac { 1 } { x ^ { 2 } }$ ?

Use a graphing calculator to approximate the real zeros. Give each zero as a decimal to the nearest tenth. - $f ( x ) = 4 x ^ { 3 } + x ^ { 2 } - 20 x - 5$

Solve the problem. - $f(x)=-(x-4)^{4}$ $Solve the problem. - f(x)=-(x-4)^{4}$

Provide an appropriate response. -A quadratic equation $f ( x ) = 0$ has a solution $x = 3$ . Its graph has vertex $( - 1 , - 16 )$ . What is the other solution of the equation?

Sketch the graph of the rational function. - $f ( x ) = \frac { ( x - 1 ) ( x - 3 ) } { ( x + 5 ) ^ { 2 } }$ $Sketch the graph of the rational function. - f ( x ) = \frac { ( x - 1 ) ( x - 3 ) } { ( x + 5 ) ^ { 2 } }$

Give all possible rational zeros for the following polynomial. - $P ( x ) = 3 x ^ { 3 } + 3 x ^ { 2 } + 2 x - 6$

Use the factor theorem to decide whether or not the second polynomial is a factor of the first. - $x ^ { 3 } - 5 x ^ { 2 } + 11 x - 15 ; x - 3$