Question 1

Divide using synthetic division.
-$$\frac { 3 x ^ { 2 } - 5 x - 28 } { x - 4 }$$

Accepted Answer

A)  $3 x + 7$ 
B)  $x + 7$ 
C)  $7 x - 4$ 
D)  $- 3 x - 7$ 
A)  $3 x + 7$ 
B)  $x + 7$ 
C)  $7 x - 4$ 
D)  $- 3 x - 7$

Question 2

Determine the constant of variation for the stated condition.
-$c$ varies jointly as a and $b$, and $c = 108$ when $a = 60$, and $b = 45$.

Accepted Answer

A)  $\mathrm { k } = \frac { 1 } { 25 }$ 
B)  $\mathrm { k } = \frac { 1 } { 45 }$ 
C)  $k = 25$ 
D)  $\mathrm { k } = 45$ 
A)  $\mathrm { k } = \frac { 1 } { 25 }$ 
B)  $\mathrm { k } = \frac { 1 } { 45 }$ 
C)  $k = 25$ 
D)  $\mathrm { k } = 45$

Question 3

Solve the problem.
-The total profit function $P ( x )$ for a company producing $x$ thousand units is given by $P ( x ) = - 2 x ^ { 2 } + 30 x - 100$. Find the values of $x$ for which the company makes a profit. [Hint: The company makes a profit when $\mathrm { P } ( \mathrm { x } ) > 0$.]

Accepted Answer

A)  $x$ is between 5 thousand units and 10 thousand units 
B)  $x$ is greater than 5 thousand units 
C)  $x$ is less than 10 thousand units 
D)  $x$ is less than 5 thousand units or greater than 10 thousand units 
A)  $x$ is between 5 thousand units and 10 thousand units 
B)  $x$ is greater than 5 thousand units 
C)  $x$ is less than 10 thousand units 
D)  $x$ is less than 5 thousand units or greater than 10 thousand units

Question 4

Use transformations of f(x) $$f ( x ) = \frac { 1 } { x } \text { or } f ( x ) = \frac { 1 } { x ^ { 2 } }$$ to graph the rational function.
-$f(x)=\frac{1}{x^{2}}+2$

Accepted Answer

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

Question 5

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for the given
function.
-$$f ( x ) = x ^ { 7 } + x ^ { 6 } + x ^ { 2 } + x + 10$$

Accepted Answer

A)  0 positive zeros, 3 or 1 negative zeros 
B)  0 positive zeros, 0 negative zeros 
C)  0 positive zeros, 2 or 0 negative zeros 
D)  0 positive zeros, 1 negative zero 
A)  0 positive zeros, 3 or 1 negative zeros 
B)  0 positive zeros, 0 negative zeros 
C)  0 positive zeros, 2 or 0 negative zeros 
D)  0 positive zeros, 1 negative zero

Question 6

Write an equation that expresses the relationship. Use k as the constant of variation.
-The weight of a body above the surface of the earth is inversely proportional to the square of its distance from the center of the earth. What is the effect on the weight when the distance is multiplied by 2?

Accepted Answer

A)  The weight is divided by 4 
B)  The weight is divided by 2 
C)  The weight is multiplied by 4 
D)  The weight is multiplied by 2 
A)  The weight is divided by 4 
B)  The weight is divided by 2 
C)  The weight is multiplied by 4 
D)  The weight is multiplied by 2

Question 7

Solve Problems Involving Joint Variation
-$z$ varies jointly as $y$ and the cube of $x ; z = 80$ when $x = 2$ and $y = - 5$

Accepted Answer

A)  $y = - 2 x ^ { 3 } y$ 
B)  $y = - 2 x y ^ { 3 }$ 
C)  $y = 2 x ^ { 3 } y$ 
D)  $y = 2 x y ^ { 3 }$ 
A)  $y = - 2 x ^ { 3 } y$ 
B)  $y = - 2 x y ^ { 3 }$ 
C)  $y = 2 x ^ { 3 } y$ 
D)  $y = 2 x y ^ { 3 }$

Question 8

Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational
roots.
-$$x ^ { 3 } + 3 x ^ { 2 } + 4 x - 8 = 0$$

Accepted Answer

A)  $\{ 1 , - 2 + 2 i , - 2 - 2 i \}$ 
B)  $\{ 1,2 + 2 i , 2 - 2 i \}$ 
C)  $\{ - 1 , - 2 + \sqrt { 5 } , - 4 - \sqrt { 5 } \}$ 
D)  $\{ 1,2 + \sqrt { 5 } , 2 - \sqrt { 5 } \}$ 
A)  $\{ 1 , - 2 + 2 i , - 2 - 2 i \}$ 
B)  $\{ 1,2 + 2 i , 2 - 2 i \}$ 
C)  $\{ - 1 , - 2 + \sqrt { 5 } , - 4 - \sqrt { 5 } \}$ 
D)  $\{ 1,2 + \sqrt { 5 } , 2 - \sqrt { 5 } \}$

Question 9

Find a rational zero of the polynomial function and use it to find all the zeros of the function.
-$$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } + 5 x - 26$$

Accepted Answer

A)  $\{ 2 , - 2 + 3 i , - 2 - 3 i \}$ 
B)  $\{ 2,3 + 2 i , 3 - 2 i \}$ 
C)  $\{ - 2 , - 2 + \sqrt { 5 } , - 4 - \sqrt { 5 } \}$ 
D)  $\{ 2,3 + \sqrt { 5 } , 3 - \sqrt { 5 } \}$ 
A)  $\{ 2 , - 2 + 3 i , - 2 - 3 i \}$ 
B)  $\{ 2,3 + 2 i , 3 - 2 i \}$ 
C)  $\{ - 2 , - 2 + \sqrt { 5 } , - 4 - \sqrt { 5 } \}$ 
D)  $\{ 2,3 + \sqrt { 5 } , 3 - \sqrt { 5 } \}$

Question 10

Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval
notation.
-$$( 3 x - 5 ) ( x + 1 ) \leq 0$$

Accepted Answer

A)  $\left[ - 1 , \frac { 5 } { 3 } \right]$
  
B)  $( - \infty , - 1 ] \cup \left[ \frac { 5 } { 3 } , \infty \right]$
  
C)  $\left( - \infty , \frac { 5 } { 3 } \right]$
  
D)  $[ - 1 , \infty )$
  
A)  $\left[ - 1 , \frac { 5 } { 3 } \right]$
  
B)  $( - \infty , - 1 ] \cup \left[ \frac { 5 } { 3 } , \infty \right]$
  
C)  $\left( - \infty , \frac { 5 } { 3 } \right]$
  
D)  $[ - 1 , \infty )$

Question 11

Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval
notation.
-$$( x + 2 ) ( x - 2 ) \leq 0$$

Accepted Answer

A)  $[ - 2,2 ]$
  
B)  $( - 2,2 )$
  
C)  $( - \infty , - 2 ] \cup [ 2 , \infty )$
  
D)  $( - \infty , - 2 ) \cup ( 2 , \infty )$
  
A)  $[ - 2,2 ]$
  
B)  $( - 2,2 )$
  
C)  $( - \infty , - 2 ] \cup [ 2 , \infty )$
  
D)  $( - \infty , - 2 ) \cup ( 2 , \infty )$

Question 12

Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational
roots.
-$$3 x ^ { 3 } - 19 x ^ { 2 } + 30 x - 8 = 0$$

Accepted Answer

A)  $\left\{ \frac { 1 } { 3 } , 2,4 \right\}$ 
B)  $\left\{ - \frac { 1 } { 3 } , 2 , - 4 \right\}$ 
C)  $\left\{ \frac { 4 } { 3 } , 1,2 \right\}$ 
D)  $\left\{ - \frac { 4 } { 3 } , 1 , - 2 \right\}$ 
A)  $\left\{ \frac { 1 } { 3 } , 2,4 \right\}$ 
B)  $\left\{ - \frac { 1 } { 3 } , 2 , - 4 \right\}$ 
C)  $\left\{ \frac { 4 } { 3 } , 1,2 \right\}$ 
D)  $\left\{ - \frac { 4 } { 3 } , 1 , - 2 \right\}$

Question 13

Find the indicated intercept(s) of the graph of the function.
-Is there origin symmetry for the rational function $f ( x ) = \frac { 5 x ^ { 2 } - 1 } { - 5 x ^ { 2 } + 2 }$ ?

Accepted Answer

A)  Yes 
B)  $\mathrm { No }$ 
A)  Yes 
B)  $\mathrm { No }$

Question 14

Write an equation that expresses the relationship. Use k as the constant of variation.
-c varies directly as $\mathrm { v }$.

Accepted Answer

A)  $\mathrm { c } = \mathrm { kV }$ 
B)  $\mathrm { c } = \frac { \mathrm { k } } { \mathrm { v } }$ 
C)  $\mathrm { k } = \mathrm { cV }$ 
D)  $v = \frac { k } { c }$ 
A)  $\mathrm { c } = \mathrm { kV }$ 
B)  $\mathrm { c } = \frac { \mathrm { k } } { \mathrm { v } }$ 
C)  $\mathrm { k } = \mathrm { cV }$ 
D)  $v = \frac { k } { c }$

Question 15

Solve the problem.
-A number minus the product of 36 and its reciprocal is less than zero. Find the numbers which satisfy this condition.

Accepted Answer

A)  any number less than $- 6$ or between 0 and 6 
B)  any number between 0 and 6 
C)  any number between $- 6$ and 6 
D)  any number less than 6 
A)  any number less than $- 6$ or between 0 and 6 
B)  any number between 0 and 6 
C)  any number between $- 6$ and 6 
D)  any number less than 6

Question 16

Use the Rational Zero Theorem to list all possible rational zeros for the given function.
-$$f ( x ) = x ^ { 5 } - 2 x ^ { 2 } + 3 x + 15$$

Accepted Answer

A)  $\pm 1 , \pm 5 , \pm 3 , \pm 15$ 
B)  $\pm 1 , \pm \frac { 1 } { 5 } , \pm \frac { 1 } { 3 } , \pm \frac { 1 } { 15 }$ 
C)  $\pm 1 , \pm \frac { 1 } { 5 } , \pm \frac { 1 } { 3 } , \pm \frac { 1 } { 15 } , \pm 5 , \pm 3 , \pm 15$ 
D)  $\pm 1 , \pm 5 , \pm 3$ 
A)  $\pm 1 , \pm 5 , \pm 3 , \pm 15$ 
B)  $\pm 1 , \pm \frac { 1 } { 5 } , \pm \frac { 1 } { 3 } , \pm \frac { 1 } { 15 }$ 
C)  $\pm 1 , \pm \frac { 1 } { 5 } , \pm \frac { 1 } { 3 } , \pm \frac { 1 } { 15 } , \pm 5 , \pm 3 , \pm 15$ 
D)  $\pm 1 , \pm 5 , \pm 3$

Question 17

Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval
notation.
-$$\frac { 4 x + 5 } { 12 - 4 x } \geq 0$$

Accepted Answer

A)  $\left[ - \frac { 5 } { 4 } , 3 \right]$
  
B)  $\left[ - \infty , - \frac { 5 } { 4 } \right]$ or $( 3 , \infty )$
  
C)  $\left[ - \frac { 5 } { 4 } , 3 \right]$
  
D)  $\left[ - \frac { 5 } { 4 } , \infty \right]$
  
A)  $\left[ - \frac { 5 } { 4 } , 3 \right]$
  
B)  $\left[ - \infty , - \frac { 5 } { 4 } \right]$ or $( 3 , \infty )$
  
C)  $\left[ - \frac { 5 } { 4 } , 3 \right]$
  
D)  $\left[ - \frac { 5 } { 4 } , \infty \right]$

Question 18

Use the graph of the rational function shown to complete the statement.
- 
As $x \rightarrow 0 ^ { + } , f ( x ) \rightarrow$ ?

Accepted Answer

A)  $+ \infty$ 
B)  $- \infty$ 
C)  1 
D)  0 
A)  $+ \infty$ 
B)  $- \infty$ 
C)  1 
D)  0

Question 19

Write an equation that expresses the relationship. Use k as the constant of variation.
-$g$ varies inversely as $y$.

Accepted Answer

A)  $g = \frac { k } { y }$ 
B)  $g = \frac { y } { k }$ 
C)  $g = k y$ 
D)  $\mathrm { kg } = \mathrm { y }$ 
A)  $g = \frac { k } { y }$ 
B)  $g = \frac { y } { k }$ 
C)  $g = k y$ 
D)  $\mathrm { kg } = \mathrm { y }$

Question 20

Determine whether the function is a polynomial function.
-$$f ( x ) = - 12 x ^ { 6 } - 3 x + \frac { 6 } { x }$$

Accepted Answer

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

Divide using synthetic division. - $\frac { 3 x ^ { 2 } - 5 x - 28 } { x - 4 }$

Determine the constant of variation for the stated condition. - $c$ varies jointly as a and $b$ , and $c = 108$ when $a = 60$ , and $b = 45$ .

Solve the problem. -The total profit function $P ( x )$ for a company producing $x$ thousand units is given by $P ( x ) = - 2 x ^ { 2 } + 30 x - 100$ . Find the values of $x$ for which the company makes a profit. [Hint: The company makes a profit when $\mathrm { P } ( \mathrm { x } ) > 0$ .]

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for the given function. - $f ( x ) = x ^ { 7 } + x ^ { 6 } + x ^ { 2 } + x + 10$

Write an equation that expresses the relationship. Use k as the constant of variation. -The weight of a body above the surface of the earth is inversely proportional to the square of its distance from the center of the earth. What is the effect on the weight when the distance is multiplied by 2?

Solve Problems Involving Joint Variation - $z$ varies jointly as $y$ and the cube of $x ; z = 80$ when $x = 2$ and $y = - 5$

Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots. - $x ^ { 3 } + 3 x ^ { 2 } + 4 x - 8 = 0$

Find a rational zero of the polynomial function and use it to find all the zeros of the function. - $f ( x ) = x ^ { 3 } + 2 x ^ { 2 } + 5 x - 26$

Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots. - $3 x ^ { 3 } - 19 x ^ { 2 } + 30 x - 8 = 0$

Find the indicated intercept(s) of the graph of the function. -Is there origin symmetry for the rational function $f ( x ) = \frac { 5 x ^ { 2 } - 1 } { - 5 x ^ { 2 } + 2 }$ ?

Write an equation that expresses the relationship. Use k as the constant of variation. -c varies directly as $\mathrm { v }$ .

Solve the problem. -A number minus the product of 36 and its reciprocal is less than zero. Find the numbers which satisfy this condition.

Use the Rational Zero Theorem to list all possible rational zeros for the given function. - $f ( x ) = x ^ { 5 } - 2 x ^ { 2 } + 3 x + 15$

Use the graph of the rational function shown to complete the statement. - $Use the graph of the rational function shown to complete the statement. - As x \rightarrow 0 ^ { + } , f ( x ) \rightarrow ?$ As $x \rightarrow 0 ^ { + } , f ( x ) \rightarrow$ ?

Write an equation that expresses the relationship. Use k as the constant of variation. - $g$ varies inversely as $y$ .

Determine whether the function is a polynomial function. - $f ( x ) = - 12 x ^ { 6 } - 3 x + \frac { 6 } { x }$

Equations and Inequalities

Functions and Graphs

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Systems of Equations and Inequalities

Matrices and Determinants

Conic Sections and Analytic Geometry

Sequences, Induction, and Probability

Prerequisites: Fundamental Concepts of Algebra

Filters

Exam 3: Polynomial and Rational Functions

Divide using synthetic division. - 3x2−5x−28x−4\frac { 3 x ^ { 2 } - 5 x - 28 } { x - 4 }x−43x2−5x−28​

Determine the constant of variation for the stated condition. - ccc varies jointly as a and bbb , and c=108c = 108c=108 when a=60a = 60a=60 , and b=45b = 45b=45 .

Use transformations of f(x) f(x)=1x or f(x)=1x2f ( x ) = \frac { 1 } { x } \text { or } f ( x ) = \frac { 1 } { x ^ { 2 } }f(x)=x1​ or f(x)=x21​ to graph the rational function. - f(x)=1x2+2f(x)=\frac{1}{x^{2}}+2f(x)=x21​+2

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for the given function. - f(x)=x7+x6+x2+x+10f ( x ) = x ^ { 7 } + x ^ { 6 } + x ^ { 2 } + x + 10f(x)=x7+x6+x2+x+10

Write an equation that expresses the relationship. Use k as the constant of variation. -The weight of a body above the surface of the earth is inversely proportional to the square of its distance from the center of the earth. What is the effect on the weight when the distance is multiplied by 2?

Solve Problems Involving Joint Variation - zzz varies jointly as yyy and the cube of x;z=80x ; z = 80x;z=80 when x=2x = 2x=2 and y=−5y = - 5y=−5

Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots. - x3+3x2+4x−8=0x ^ { 3 } + 3 x ^ { 2 } + 4 x - 8 = 0x3+3x2+4x−8=0

Find a rational zero of the polynomial function and use it to find all the zeros of the function. - f(x)=x3+2x2+5x−26f ( x ) = x ^ { 3 } + 2 x ^ { 2 } + 5 x - 26f(x)=x3+2x2+5x−26

Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval notation. - (3x−5)(x+1)≤0( 3 x - 5 ) ( x + 1 ) \leq 0(3x−5)(x+1)≤0

Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval notation. - (x+2)(x−2)≤0( x + 2 ) ( x - 2 ) \leq 0(x+2)(x−2)≤0

Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots. - 3x3−19x2+30x−8=03 x ^ { 3 } - 19 x ^ { 2 } + 30 x - 8 = 03x3−19x2+30x−8=0

Find the indicated intercept(s) of the graph of the function. -Is there origin symmetry for the rational function f(x)=5x2−1−5x2+2f ( x ) = \frac { 5 x ^ { 2 } - 1 } { - 5 x ^ { 2 } + 2 }f(x)=−5x2+25x2−1​ ?

Write an equation that expresses the relationship. Use k as the constant of variation. -c varies directly as v\mathrm { v }v .