Question 1

Find the vertical asymptotes of the rational function.
-$$f ( x ) = \frac { x ( x - 1 ) } { 49 x ^ { 2 } + 84 x + 20 }$$

Accepted Answer

A)  $x = \frac { 2 } { 7 } , x = \frac { 10 } { 7 }$ 
B)  $x = - \frac { 1 } { 25 } , x = - \frac { 3 } { 25 }$ 
C) $ \mathrm { x } = - \frac { 10 } { 49 } , x = \frac { 30 } { 49 }$ 
D)  $x = - \frac { 1 } { 5 } , x = - \frac { 3 } { 5 }$ 
A)  $x = \frac { 2 } { 7 } , x = \frac { 10 } { 7 }$ 
B)  $x = - \frac { 1 } { 25 } , x = - \frac { 3 } { 25 }$ 
C) $ \mathrm { x } = - \frac { 10 } { 49 } , x = \frac { 30 } { 49 }$ 
D)  $x = - \frac { 1 } { 5 } , x = - \frac { 3 } { 5 }$

Question 2

Use transformations of the graph o to graph the function. $$y = x ^ { 4 } \text { or } y = x ^ { 5 }$$
-$$f ( x ) = \frac { 1 } { 2 } ( x + 4 ) ^ { 4 } + 3$$

Accepted Answer

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

Question 3

Use the Factor Theorem to determine whether x - c is a factor of f(x).
-$7 x ^ { 3 } + 17 x ^ { 2 } - 11 x + 3 ; x + 3$

Accepted Answer

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

Question 4

Find the domain of the rational function.
-$$h ( x ) = \frac { x + 2 } { x ^ { 2 } - 36 }$$

Accepted Answer

A)  $\{ x \mid x \neq - 6 , x \neq 6 \}$ 
B)  $\{ x \mid x \neq - 6 , x \neq 6 , x \neq - 2 \}$ 
C)  all real numbers 
D)  $\{ x \mid x \neq 0 , x \neq 36 \}$ 
A)  $\{ x \mid x \neq - 6 , x \neq 6 \}$ 
B)  $\{ x \mid x \neq - 6 , x \neq 6 , x \neq - 2 \}$ 
C)  all real numbers 
D)  $\{ x \mid x \neq 0 , x \neq 36 \}$

Question 5

Use the Factor Theorem to determine whether x - c is a factor of f(x).
-$f ( x ) = 15 x ^ { 3 } + 49 x ^ { 2 } - 49 x - 99 ; x - \frac { 11 } { 3 }$

Accepted Answer

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

Question 6

Use the Factor Theorem to determine whether x - c is a factor of f(x).
-$f ( x ) = x ^ { 4 } - 12 x ^ { 2 } - 64 ; x - 4$

Accepted Answer

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

Question 7

Use transformations of the graph o to graph the function. $$y = x ^ { 4 } \text { or } y = x ^ { 5 }$$
-$f(x)=-2 x^{4}$

Accepted Answer

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

Question 8

Give the equation of the oblique asymptote, if any, of the function.
-$$h ( x ) = \frac { 7 x ^ { 2 } - 9 x - 4 } { 5 x ^ { 2 } - 2 x + 5 }$$

Accepted Answer

A)  $y = \frac { 7 } { 5 } x$ 
B)  $y = x + \frac { 7 } { 5 }$ 
C)  $y = \frac { 7 } { 5 }$ 
D)  no oblique asymptote 
A)  $y = \frac { 7 } { 5 } x$ 
B)  $y = x + \frac { 7 } { 5 }$ 
C)  $y = \frac { 7 } { 5 }$ 
D)  no oblique asymptote

Question 9

Find the vertical asymptotes of the rational function.
-$$g ( x ) = \frac { 6 x } { x + 5 }$$

Accepted Answer

A)  $x = - 5$ 
B)  $x = 5$ 
C)  none 
D)  $x = 6$ 
A)  $x = - 5$ 
B)  $x = 5$ 
C)  none 
D)  $x = 6$

Question 10

Find the vertical asymptotes of the rational function.
-$$R ( x ) = \frac { - 3 x ^ { 2 } } { x ^ { 2 } + 9 x - 36 }$$

Accepted Answer

A)  $x = - 36$ 
B)  $x = - 12 , x = 3 , x = - 3$ 
C)  $x = - 12 , x = 3$ 
D)  $x = 12 , x = - 3$ 
A)  $x = - 36$ 
B)  $x = - 12 , x = 3 , x = - 3$ 
C)  $x = - 12 , x = 3$ 
D)  $x = 12 , x = - 3$

Question 11

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the
zeros to factor f over the real numbers.
-$$f ( x ) = 5 x ^ { 4 } - 8 x ^ { 3 } + 23 x ^ { 2 } - 32 x + 12$$

Accepted Answer

A)  $- 4 , - 1,1 , \frac { 3 } { 5 } ; f ( x ) = ( x - 1 ) ( 5 x - 3 ) ( x + 1 ) ( x + 4 )$ 
B)  $1 , \frac { 3 } { 5 } ; f ( x ) = ( x - 1 ) ( 5 x - 3 ) \left( x ^ { 2 } + 4 \right)$ 
C) $4 , \frac { 3 } { 5 } ; f ( x ) = ( x - 4 ) ( 5 x - 3 ) \left( x ^ { 2 } + 1 \right)$ 
D)  $- 4 , - 1,1 , - \frac { 3 } { 5 } ; f ( x ) = ( x - 1 ) ( 5 x + 3 ) ( x + 1 ) ( x + 4 )$ 
A)  $- 4 , - 1,1 , \frac { 3 } { 5 } ; f ( x ) = ( x - 1 ) ( 5 x - 3 ) ( x + 1 ) ( x + 4 )$ 
B)  $1 , \frac { 3 } { 5 } ; f ( x ) = ( x - 1 ) ( 5 x - 3 ) \left( x ^ { 2 } + 4 \right)$ 
C) $4 , \frac { 3 } { 5 } ; f ( x ) = ( x - 4 ) ( 5 x - 3 ) \left( x ^ { 2 } + 1 \right)$ 
D)  $- 4 , - 1,1 , - \frac { 3 } { 5 } ; f ( x ) = ( x - 1 ) ( 5 x + 3 ) ( x + 1 ) ( x + 4 )$

Question 12

Use the graph to determine the domain and range of the function.
-

Accepted Answer

A)  domain: $\{ x \mid x > 0 \}$
range: $\{ y \mid y \neq 4 \}$ 
B)  domain: $\{ x \mid x \neq 4 \}$
range: $\{ y \mid y > 0 \}$ 
C)  domain: $\{ x \mid x \neq 4 \}$
range: $\{ y \mid y \geq 0 \}$ 
D)  domain: $\{ x \mid x \geq 0 \}$
range: $\{ y \mid y \neq 4 \}$ 
A)  domain: $\{ x \mid x > 0 \}$
range: $\{ y \mid y \neq 4 \}$ 
B)  domain: $\{ x \mid x \neq 4 \}$
range: $\{ y \mid y > 0 \}$ 
C)  domain: $\{ x \mid x \neq 4 \}$
range: $\{ y \mid y \geq 0 \}$ 
D)  domain: $\{ x \mid x \geq 0 \}$
range: $\{ y \mid y \neq 4 \}$

Question 13

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
-$f ( x ) = 1 + \frac { 6 } { x }$

Accepted Answer

A)  Yes; degree 0 
B)  Yes; degree 6 
C)  Yes; degree 1 
D)  No; x is raised to a negative power 
A)  Yes; degree 0 
B)  Yes; degree 6 
C)  Yes; degree 1 
D)  No; x is raised to a negative power

Question 14

Find all zeros of the function and write the polynomial as a product of linear factors.
-$f ( x ) = x ^ { 4 } + 5 x ^ { 3 } + 15 x ^ { 2 } + 45 x + 54$

Accepted Answer

A)  $f ( x ) = ( x + 3 ) ( x + 2 ) ( x - 3 i ) ( x + 3 i )$ 
B)  $f ( x ) = ( x - 3 ) ( x + 2 ) ( x - 3 ) ( x + 3 )$ 
C)  $f ( x ) = ( x - 1 ) ( x - 6 ) ( x - 3 i ) ( x + 3 i )$ 
D)  $f ( x ) = ( x - i \sqrt { 6 } ) ( x + i \sqrt { 6 } ) ( x - 3 ) ( x + 3 )$ 
A)  $f ( x ) = ( x + 3 ) ( x + 2 ) ( x - 3 i ) ( x + 3 i )$ 
B)  $f ( x ) = ( x - 3 ) ( x + 2 ) ( x - 3 ) ( x + 3 )$ 
C)  $f ( x ) = ( x - 1 ) ( x - 6 ) ( x - 3 i ) ( x + 3 i )$ 
D)  $f ( x ) = ( x - i \sqrt { 6 } ) ( x + i \sqrt { 6 } ) ( x - 3 ) ( x + 3 )$

Question 15

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at
each x -intercept.
-$$f ( x ) = \left( x + \frac { 1 } { 4 } \right) ^ { 2 } \left( x ^ { 2 } + 7 \right) ^ { 3 }$$

Accepted Answer

A)  $- \frac { 1 } { 4 }$, multiplicity 2, touches $x$-axis 
B)  $- \frac { 1 } { 4 }$, multiplicity 2, touches $x$-axis; $- 7$, multiplicity 3 , crosses $x$-axis 
C)  $- \frac { 1 } { 4 }$, multiplicity 2, crosses x-axis 
D)  $\frac { 1 } { 4 }$, multiplicity 2 , touches $x$-axis; 7 , multiplicity 3 , crosses $x$-axis 
A)  $- \frac { 1 } { 4 }$, multiplicity 2, touches $x$-axis 
B)  $- \frac { 1 } { 4 }$, multiplicity 2, touches $x$-axis; $- 7$, multiplicity 3 , crosses $x$-axis 
C)  $- \frac { 1 } { 4 }$, multiplicity 2, crosses x-axis 
D)  $\frac { 1 } { 4 }$, multiplicity 2 , touches $x$-axis; 7 , multiplicity 3 , crosses $x$-axis

Question 16

Find the vertical asymptotes of the rational function.
-$$f ( x ) = \frac { x - 4 } { 16 x - x ^ { 3 } }$$

Accepted Answer

A)  $x = 0 , x = 4$ 
B)  $x = 0 , x = - 4$ 
C)  $x = - 4 , x = 4$ 
D)  $x = 0 , x = - 4 , x = 4$ 
A)  $x = 0 , x = 4$ 
B)  $x = 0 , x = - 4$ 
C)  $x = - 4 , x = 4$ 
D)  $x = 0 , x = - 4 , x = 4$

Question 17

Graph the function.
-$$f ( x ) = \frac { x ^ { 2 } - 5 x + 4 } { ( x - 5 ) ^ { 2 } }$$

Accepted Answer

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

Question 18

Write the word or phrase that best completes each statement or answers the question.
Analyze the graph of the given function f as follows:
(a) Determine the end behavior: find the power function that the graph of f resembles for large values of |x|.
(b) Find the x- and y-intercepts of the graph.
(c) Determine whether the graph crosses or touches the x-axis at each x-intercept.
(d) Graph f using a graphing utility.
(e) Use the graph to determine the local maxima and local minima, if any exist. Round turning points to two decimal places.
(f) Use the information obtained in (a) - (e) to draw a complete graph of f by hand. Label all intercepts and turning points.
(g) Find the domain of f. Use the graph to find the range of f.
(h) Use the graph to determine where f is increasing and where f is decreasing.
-$$f ( x ) = ( x - 3 ) ( x - 1 ) ( x + 2 )$$

Accepted Answer

(a) For large values of @#LAT-DLM&, the graph of @#LAT-DLM&

Question 19

Give the maximum number of zeros the polynomial function may have. Use Descarte's Rule of Signs to determine how
many positive and how many negative zeros it may have.
-$$f ( x ) = x ^ { 2 } - 18$$

Accepted Answer

A)  $2 ; 0$ positive zeros; 0 negative zeros 
B)  2; 0 positive zeros; 1 negative zero 
C)  2 ; 1 positive zero; 0 negative zeros 
D)  2 ; 1 positive zero; 1 negative zero 
A)  $2 ; 0$ positive zeros; 0 negative zeros 
B)  2; 0 positive zeros; 1 negative zero 
C)  2 ; 1 positive zero; 0 negative zeros 
D)  2 ; 1 positive zero; 1 negative zero

Question 20

Give the maximum number of zeros the polynomial function may have. Use Descarte's Rule of Signs to determine how
many positive and how many negative zeros it may have.
-$$f ( x ) = x ^ { 6 } - x ^ { 5 } - x ^ { 4 } - 5 x ^ { 3 } + 2 x ^ { 2 } + 2 x + 3$$

Accepted Answer

A)  $6 ; 2$ or 0 positive zeros; 2 , or 0 negative zeros 
B)  6; 3 or 1 positive zeros; 3 or 1 negative zeros 
C)  6; 2 or 0 positive zeros; 4,2 , or 0 negative zeros 
D)  $6 ; 4,2$, or 0 positive zeros; 2 or 0 negative zeros 
A)  $6 ; 2$ or 0 positive zeros; 2 , or 0 negative zeros 
B)  6; 3 or 1 positive zeros; 3 or 1 negative zeros 
C)  6; 2 or 0 positive zeros; 4,2 , or 0 negative zeros 
D)  $6 ; 4,2$, or 0 positive zeros; 2 or 0 negative zeros

Find the vertical asymptotes of the rational function. - $f ( x ) = \frac { x ( x - 1 ) } { 49 x ^ { 2 } + 84 x + 20 }$

Use the Factor Theorem to determine whether x - c is a factor of f(x). - $7 x ^ { 3 } + 17 x ^ { 2 } - 11 x + 3 ; x + 3$

Find the domain of the rational function. - $h ( x ) = \frac { x + 2 } { x ^ { 2 } - 36 }$

Use the Factor Theorem to determine whether x - c is a factor of f(x). - $f ( x ) = 15 x ^ { 3 } + 49 x ^ { 2 } - 49 x - 99 ; x - \frac { 11 } { 3 }$

Use the Factor Theorem to determine whether x - c is a factor of f(x). - $f ( x ) = x ^ { 4 } - 12 x ^ { 2 } - 64 ; x - 4$

Use transformations of the graph o to graph the function. $y = x ^ { 4 } \text { or } y = x ^ { 5 }$ - $f(x)=-2 x^{4}$ $Use transformations of the graph o to graph the function. y = x ^ { 4 } \text { or } y = x ^ { 5 } - f(x)=-2 x^{4}$

Give the equation of the oblique asymptote, if any, of the function. - $h ( x ) = \frac { 7 x ^ { 2 } - 9 x - 4 } { 5 x ^ { 2 } - 2 x + 5 }$

Find the vertical asymptotes of the rational function. - $g ( x ) = \frac { 6 x } { x + 5 }$

Find the vertical asymptotes of the rational function. - $R ( x ) = \frac { - 3 x ^ { 2 } } { x ^ { 2 } + 9 x - 36 }$

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. - $f ( x ) = 5 x ^ { 4 } - 8 x ^ { 3 } + 23 x ^ { 2 } - 32 x + 12$

Use the graph to determine the domain and range of the function. -

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $f ( x ) = 1 + \frac { 6 } { x }$

Find all zeros of the function and write the polynomial as a product of linear factors. - $f ( x ) = x ^ { 4 } + 5 x ^ { 3 } + 15 x ^ { 2 } + 45 x + 54$

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. - $f ( x ) = \left( x + \frac { 1 } { 4 } \right) ^ { 2 } \left( x ^ { 2 } + 7 \right) ^ { 3 }$

Find the vertical asymptotes of the rational function. - $f ( x ) = \frac { x - 4 } { 16 x - x ^ { 3 } }$

Graph the function. - $f ( x ) = \frac { x ^ { 2 } - 5 x + 4 } { ( x - 5 ) ^ { 2 } }$ $Graph the function. - f ( x ) = \frac { x ^ { 2 } - 5 x + 4 } { ( x - 5 ) ^ { 2 } }$

Give the maximum number of zeros the polynomial function may have. Use Descarte's Rule of Signs to determine how many positive and how many negative zeros it may have. - $f ( x ) = x ^ { 2 } - 18$

Give the maximum number of zeros the polynomial function may have. Use Descarte's Rule of Signs to determine how many positive and how many negative zeros it may have. - $f ( x ) = x ^ { 6 } - x ^ { 5 } - x ^ { 4 } - 5 x ^ { 3 } + 2 x ^ { 2 } + 2 x + 3$

Functions and Their Graphs

Linear and Quadratic Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Exam 3: Polynomial and Rational Functions

Find the vertical asymptotes of the rational function. - f(x)=x(x−1)49x2+84x+20f ( x ) = \frac { x ( x - 1 ) } { 49 x ^ { 2 } + 84 x + 20 }f(x)=49x2+84x+20x(x−1)​

Use transformations of the graph o to graph the function. y=x4 or y=x5y = x ^ { 4 } \text { or } y = x ^ { 5 }y=x4 or y=x5 - f(x)=12(x+4)4+3f ( x ) = \frac { 1 } { 2 } ( x + 4 ) ^ { 4 } + 3f(x)=21​(x+4)4+3

Use the Factor Theorem to determine whether x - c is a factor of f(x). - 7x3+17x2−11x+3;x+37 x ^ { 3 } + 17 x ^ { 2 } - 11 x + 3 ; x + 37x3+17x2−11x+3;x+3

Find the domain of the rational function. - h(x)=x+2x2−36h ( x ) = \frac { x + 2 } { x ^ { 2 } - 36 }h(x)=x2−36x+2​

Use the Factor Theorem to determine whether x - c is a factor of f(x). - f(x)=15x3+49x2−49x−99;x−113f ( x ) = 15 x ^ { 3 } + 49 x ^ { 2 } - 49 x - 99 ; x - \frac { 11 } { 3 }f(x)=15x3+49x2−49x−99;x−311​

Use the Factor Theorem to determine whether x - c is a factor of f(x). - f(x)=x4−12x2−64;x−4f ( x ) = x ^ { 4 } - 12 x ^ { 2 } - 64 ; x - 4f(x)=x4−12x2−64;x−4

Use transformations of the graph o to graph the function. y=x4 or y=x5y = x ^ { 4 } \text { or } y = x ^ { 5 }y=x4 or y=x5 - f(x)=−2x4f(x)=-2 x^{4}f(x)=−2x4

Give the equation of the oblique asymptote, if any, of the function. - h(x)=7x2−9x−45x2−2x+5h ( x ) = \frac { 7 x ^ { 2 } - 9 x - 4 } { 5 x ^ { 2 } - 2 x + 5 }h(x)=5x2−2x+57x2−9x−4​

Find the vertical asymptotes of the rational function. - g(x)=6xx+5g ( x ) = \frac { 6 x } { x + 5 }g(x)=x+56x​

Find the vertical asymptotes of the rational function. - R(x)=−3x2x2+9x−36R ( x ) = \frac { - 3 x ^ { 2 } } { x ^ { 2 } + 9 x - 36 }R(x)=x2+9x−36−3x2​

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. - f(x)=5x4−8x3+23x2−32x+12f ( x ) = 5 x ^ { 4 } - 8 x ^ { 3 } + 23 x ^ { 2 } - 32 x + 12f(x)=5x4−8x3+23x2−32x+12

Use the graph to determine the domain and range of the function. -

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - f(x)=1+6xf ( x ) = 1 + \frac { 6 } { x }f(x)=1+x6​

Find all zeros of the function and write the polynomial as a product of linear factors. - f(x)=x4+5x3+15x2+45x+54f ( x ) = x ^ { 4 } + 5 x ^ { 3 } + 15 x ^ { 2 } + 45 x + 54f(x)=x4+5x3+15x2+45x+54

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. - f(x)=(x+14)2(x2+7)3f ( x ) = \left( x + \frac { 1 } { 4 } \right) ^ { 2 } \left( x ^ { 2 } + 7 \right) ^ { 3 }f(x)=(x+41​)2(x2+7)3

Find the vertical asymptotes of the rational function. - f(x)=x−416x−x3f ( x ) = \frac { x - 4 } { 16 x - x ^ { 3 } }f(x)=16x−x3x−4​

Graph the function. - f(x)=x2−5x+4(x−5)2f ( x ) = \frac { x ^ { 2 } - 5 x + 4 } { ( x - 5 ) ^ { 2 } }f(x)=(x−5)2x2−5x+4​

Give the maximum number of zeros the polynomial function may have. Use Descarte's Rule of Signs to determine how many positive and how many negative zeros it may have. - f(x)=x2−18f ( x ) = x ^ { 2 } - 18f(x)=x2−18

Functions and Their Graphs

Linear and Quadratic Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Find the vertical asymptotes of the rational function. - $f ( x ) = \frac { x ( x - 1 ) } { 49 x ^ { 2 } + 84 x + 20 }$

Use the Factor Theorem to determine whether x - c is a factor of f(x). - $7 x ^ { 3 } + 17 x ^ { 2 } - 11 x + 3 ; x + 3$

Find the domain of the rational function. - $h ( x ) = \frac { x + 2 } { x ^ { 2 } - 36 }$

Use the Factor Theorem to determine whether x - c is a factor of f(x). - $f ( x ) = 15 x ^ { 3 } + 49 x ^ { 2 } - 49 x - 99 ; x - \frac { 11 } { 3 }$

Use the Factor Theorem to determine whether x - c is a factor of f(x). - $f ( x ) = x ^ { 4 } - 12 x ^ { 2 } - 64 ; x - 4$

Use transformations of the graph o to graph the function. $y = x ^ { 4 } \text { or } y = x ^ { 5 }$ - $f(x)=-2 x^{4}$ $Use transformations of the graph o to graph the function. y = x ^ { 4 } \text { or } y = x ^ { 5 } - f(x)=-2 x^{4}$

Give the equation of the oblique asymptote, if any, of the function. - $h ( x ) = \frac { 7 x ^ { 2 } - 9 x - 4 } { 5 x ^ { 2 } - 2 x + 5 }$

Find the vertical asymptotes of the rational function. - $g ( x ) = \frac { 6 x } { x + 5 }$

Find the vertical asymptotes of the rational function. - $R ( x ) = \frac { - 3 x ^ { 2 } } { x ^ { 2 } + 9 x - 36 }$

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. - $f ( x ) = 5 x ^ { 4 } - 8 x ^ { 3 } + 23 x ^ { 2 } - 32 x + 12$

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $f ( x ) = 1 + \frac { 6 } { x }$

Find all zeros of the function and write the polynomial as a product of linear factors. - $f ( x ) = x ^ { 4 } + 5 x ^ { 3 } + 15 x ^ { 2 } + 45 x + 54$

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. - $f ( x ) = \left( x + \frac { 1 } { 4 } \right) ^ { 2 } \left( x ^ { 2 } + 7 \right) ^ { 3 }$

Find the vertical asymptotes of the rational function. - $f ( x ) = \frac { x - 4 } { 16 x - x ^ { 3 } }$

Graph the function. - $f ( x ) = \frac { x ^ { 2 } - 5 x + 4 } { ( x - 5 ) ^ { 2 } }$ $Graph the function. - f ( x ) = \frac { x ^ { 2 } - 5 x + 4 } { ( x - 5 ) ^ { 2 } }$

Give the maximum number of zeros the polynomial function may have. Use Descarte's Rule of Signs to determine how many positive and how many negative zeros it may have. - $f ( x ) = x ^ { 2 } - 18$