Question 1

Find the vertical asymptotes of the rational function.
-$$f ( x ) = \frac { - 2 x ( x + 2 ) } { 4 x ^ { 2 } - 5 x - 9 }$$

Accepted Answer

A)  $x = \frac { 4 } { 9 } , x = - 1$ 
B)  $x = - \frac { 9 } { 4 } , x = 1$ 
C)  $x = \frac { 9 } { 4 } , x = - 1$ 
D)  $x = - \frac { 4 } { 9 } , x = 1$ 
A)  $x = \frac { 4 } { 9 } , x = - 1$ 
B)  $x = - \frac { 9 } { 4 } , x = 1$ 
C)  $x = \frac { 9 } { 4 } , x = - 1$ 
D)  $x = - \frac { 4 } { 9 } , x = 1$

Question 2

Solve the equation in the real number system.
-$x ^ { 3 } + 5 x ^ { 2 } + 2 x - 8 = 0$

Accepted Answer

A)  {2, 4} 
B)  {-4, -2, 1} 
C)  {-4, -2} 
D)  {-1, 2, 4} 
A)  {2, 4} 
B)  {-4, -2, 1} 
C)  {-4, -2} 
D)  {-1, 2, 4}

Question 3

Find the indicated intercept(s) of the graph of the function.2133:2139
-$$x \text {-intercepts of } f ( x ) = \frac { 5 } { x ^ { 2 } - x - 12 }$$

Accepted Answer

A)  (5, 0) 
B)  (-4, 0), (3, 0) 
C)  (-3, 0), (4, 0) 
D)  none 
A)  (5, 0) 
B)  (-4, 0), (3, 0) 
C)  (-3, 0), (4, 0) 
D)  none

Question 4

Use the given zero to find the remaining zeros of the function.
-$f ( x ) = x ^ { 4 } - 45 x ^ { 2 } - 196 ; \text { zero: } - 2 i$

Accepted Answer

A)  2i, 7i, -7i 
B)  2i, 7, -7 
C)  2i, 14i, -14i 
D)  2i, 14, -14 
A)  2i, 7i, -7i 
B)  2i, 7, -7 
C)  2i, 14i, -14i 
D)  2i, 14, -14

Question 5

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places.
-$\mathrm { x } ^ { 3 } - 8 \mathrm { x } - 3 = 0 ; - 1 \leq \mathrm { r } \leq 0$

Accepted Answer

The answer of The equation has a solution r in...

Question 6

Give the equation of the horizontal asymptote, if any, of the function.
-$$f ( x ) = \frac { x ^ { 2 } - 7 } { 49 x - x ^ { 4 } }$$

Accepted Answer

A)  no horizontal asymptotes 
B)  $y = - 1$ 
C)  $y = - 7 , y = 7$ 
D)  $y = 0$ 
A)  no horizontal asymptotes 
B)  $y = - 1$ 
C)  $y = - 7 , y = 7$ 
D)  $y = 0$

Question 7

Solve the problem.
-$$x ^ { 2 } - 5 x \leq 0$$

Accepted Answer

A)  $( - \infty , - 5 ]$ or $[ 0 , \infty )$ 
B)  $[ - 5,0 ]$ 
C)  $( - \infty , 0 ]$ or $[ 5 , \infty )$ 
D)  $[ 0,5 ]$ 
A)  $( - \infty , - 5 ]$ or $[ 0 , \infty )$ 
B)  $[ - 5,0 ]$ 
C)  $( - \infty , 0 ]$ or $[ 5 , \infty )$ 
D)  $[ 0,5 ]$

Question 8

Solve the problem.
-$$x ^ { 3 } \geq 125$$

Accepted Answer

A)  $[ - 5,5 ]$ 
B)  $( - \infty , - 5 ]$ or $[ 5 , \infty )$ 
C)  $[ 5 , \infty )$ 
D)  $( - \infty , 5 ]$ 
A)  $[ - 5,5 ]$ 
B)  $( - \infty , - 5 ]$ or $[ 5 , \infty )$ 
C)  $[ 5 , \infty )$ 
D)  $( - \infty , 5 ]$

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

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
-$f ( x ) = \frac { 6 } { 5 } - \frac { 1 } { 5 } x$

Accepted Answer

A)  Yes; degree 0 
B)  Yes; degree 5 
C)  No; x has a fractional coefficient 
D)  Yes; degree 1 
A)  Yes; degree 0 
B)  Yes; degree 5 
C)  No; x has a fractional coefficient 
D)  Yes; degree 1

Question 11

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at
each x -intercept.
-$f ( x ) = 3 \left( x ^ { 2 } + 5 \right) ( x + 6 ) ^ { 2 }$

Accepted Answer

A)  -6, multiplicity 2, touches x-axis 
B)  -5, multiplicity 1, touches x-axis; -6, multiplicity 2, crosses x-axis 
C)  -6, multiplicity 2, crosses x-axis 
D)  -5, multiplicity 1, crosses x-axis; -6, multiplicity 2, touches x-axis 
A)  -6, multiplicity 2, touches x-axis 
B)  -5, multiplicity 1, touches x-axis; -6, multiplicity 2, crosses x-axis 
C)  -6, multiplicity 2, crosses x-axis 
D)  -5, multiplicity 1, crosses x-axis; -6, multiplicity 2, touches x-axis

Question 12

Find the x- and y-intercepts of f.
-$f ( x ) = - x ^ { 2 } ( x + 5 ) \left( x ^ { 2 } - 1 \right)$

Accepted Answer

A)  x-intercepts: -5, -1, 0, 1; y-intercept: 0 
B)  x-intercepts: -5, 0, 1; y-intercept: -5 
C)  x-intercepts: -1, 0, 1, 5; y-intercept: 0 
D)  x-intercepts: -5, -1, 0, 1; y-intercept: -5 
A)  x-intercepts: -5, -1, 0, 1; y-intercept: 0 
B)  x-intercepts: -5, 0, 1; y-intercept: -5 
C)  x-intercepts: -1, 0, 1, 5; y-intercept: 0 
D)  x-intercepts: -5, -1, 0, 1; y-intercept: -5

Question 13

Graph the function using transformations.
-$$f ( x ) = \frac { - 2 } { x + 1 }$$
 
 A)

Accepted Answer

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

Question 14

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.
-$$f ( x ) = 4 x ^ { 4 } - 8 x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1$$

Accepted Answer

A)  $- 1,1 ; f ( x ) = ( x - 1 ) ( x + 1 ) \left( 4 x ^ { 2 } + 1 \right)$ 
B)  no real roots; $\mathrm { f } ( \mathrm { x } ) = \left( \mathrm { x } ^ { 2 } + 1 \right) \left( 4 \mathrm { x } ^ { 2 } + 1 \right)$ 
C)  $- 1$, multiplicity 2; $f ( x ) = ( x + 1 ) ^ { 2 } \left( 4 x ^ { 2 } + 1 \right)$ 
D)  1 , multiplicity 2; $\mathrm { f } ( \mathrm { x } ) = ( \mathrm { x } - 1 ) ^ { 2 } \left( 4 \mathrm { x } ^ { 2 } + 1 \right)$ 
A)  $- 1,1 ; f ( x ) = ( x - 1 ) ( x + 1 ) \left( 4 x ^ { 2 } + 1 \right)$ 
B)  no real roots; $\mathrm { f } ( \mathrm { x } ) = \left( \mathrm { x } ^ { 2 } + 1 \right) \left( 4 \mathrm { x } ^ { 2 } + 1 \right)$ 
C)  $- 1$, multiplicity 2; $f ( x ) = ( x + 1 ) ^ { 2 } \left( 4 x ^ { 2 } + 1 \right)$ 
D)  1 , multiplicity 2; $\mathrm { f } ( \mathrm { x } ) = ( \mathrm { x } - 1 ) ^ { 2 } \left( 4 \mathrm { x } ^ { 2 } + 1 \right)$

Question 15

Graph the function using transformations.
-$$f ( x ) = 6 - \frac { 1 } { ( x + 3 ) ^ { 2 } }$$
 
 A)

Accepted Answer

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

Question 16

Give the equation of the horizontal asymptote, if any, of the function.
-$$h ( x ) = \frac { 7 x ^ { 2 } - 5 x - 4 } { 8 x ^ { 2 } - 8 x + 9 }$$

Accepted Answer

A)  $y = 0$ 
B)  $y = \frac { 7 } { 8 }$ 
C)  $y = \frac { 5 } { 8 }$ 
D)  no horizontal asymptotes 
A)  $y = 0$ 
B)  $y = \frac { 7 } { 8 }$ 
C)  $y = \frac { 5 } { 8 }$ 
D)  no horizontal asymptotes

Question 17

Use the Factor Theorem to determine whether x - c is a factor of f. If it is, write f in factored form, that is, write f in the
form f(x) = (x - c)(quotient).
-$$f ( x ) = 5 x ^ { 4 } - 7 x ^ { 3 } + 17 x ^ { 2 } - 21 x + 6 ; c = 1$$

Accepted Answer

A)  Yes; $f ( x ) = ( x - 1 ) \left( 5 x ^ { 3 } + 3 x ^ { 2 } + 15 x - 6 \right)$ 
B)  Yes; $f ( x ) = ( x - 1 ) \left( 5 x ^ { 3 } - 2 x ^ { 2 } - 16 x - 6 \right)$ 
C)  Yes; $f ( x ) = ( x - 1 ) \left( 5 x ^ { 3 } - 2 x ^ { 2 } + 15 x - 6 \right)$ 
D)  No 
A)  Yes; $f ( x ) = ( x - 1 ) \left( 5 x ^ { 3 } + 3 x ^ { 2 } + 15 x - 6 \right)$ 
B)  Yes; $f ( x ) = ( x - 1 ) \left( 5 x ^ { 3 } - 2 x ^ { 2 } - 16 x - 6 \right)$ 
C)  Yes; $f ( x ) = ( x - 1 ) \left( 5 x ^ { 3 } - 2 x ^ { 2 } + 15 x - 6 \right)$ 
D)  No

Question 18

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places.
-$x ^ { 4 } - x ^ { 3 } - 7 x ^ { 2 } + 5 x + 10 = 0 ; 2 < r \leq 3$

Accepted Answer

The answer of The equation has a solution r in...

Question 19

Find the x- and y-intercepts of f.
-$f ( x ) = 5 x - x ^ { 3 }$

Accepted Answer

A)  x-intercepts: $0 , \sqrt { 5 } , - \sqrt { 5 } ; y$-intercept: 0 
B)  x-intercepts: $0 , \sqrt { 5 } , - \sqrt { 5 } ; y$-intercept: 5 
C)  $x$-intercepts: $0 , - 5 ; y$-intercept: 5 
D)  $x$-intercepts: $0 , - 5 ; y$-intercept: 0 
A)  x-intercepts: $0 , \sqrt { 5 } , - \sqrt { 5 } ; y$-intercept: 0 
B)  x-intercepts: $0 , \sqrt { 5 } , - \sqrt { 5 } ; y$-intercept: 5 
C)  $x$-intercepts: $0 , - 5 ; y$-intercept: 5 
D)  $x$-intercepts: $0 , - 5 ; y$-intercept: 0

Question 20

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
-$$f ( x ) = - 15 x ^ { 4 } + \pi x ^ { 3 } + 1$$

Accepted Answer

A)  Yes; degree 8 
B)  Yes; degree 7 
C)  No; $x ^ { 3 }$ has a non-integer coefficient 
D)  Yes; degree 4 
A)  Yes; degree 8 
B)  Yes; degree 7 
C)  No; $x ^ { 3 }$ has a non-integer coefficient 
D)  Yes; degree 4

Find the vertical asymptotes of the rational function. - $f ( x ) = \frac { - 2 x ( x + 2 ) } { 4 x ^ { 2 } - 5 x - 9 }$

Solve the equation in the real number system. - $x ^ { 3 } + 5 x ^ { 2 } + 2 x - 8 = 0$

Find the indicated intercept(s) of the graph of the function.2133:2139 - $x \text {-intercepts of } f ( x ) = \frac { 5 } { x ^ { 2 } - x - 12 }$

Use the given zero to find the remaining zeros of the function. - $f ( x ) = x ^ { 4 } - 45 x ^ { 2 } - 196 ; \text { zero: } - 2 i$

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. - $\mathrm { x } ^ { 3 } - 8 \mathrm { x } - 3 = 0 ; - 1 \leq \mathrm { r } \leq 0$

Give the equation of the horizontal asymptote, if any, of the function. - $f ( x ) = \frac { x ^ { 2 } - 7 } { 49 x - x ^ { 4 } }$

Solve the problem. - $x ^ { 2 } - 5 x \leq 0$

Solve the problem. - $x ^ { 3 } \geq 125$

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

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $f ( x ) = \frac { 6 } { 5 } - \frac { 1 } { 5 } x$

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. - $f ( x ) = 3 \left( x ^ { 2 } + 5 \right) ( x + 6 ) ^ { 2 }$

Find the x- and y-intercepts of f. - $f ( x ) = - x ^ { 2 } ( x + 5 ) \left( x ^ { 2 } - 1 \right)$

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. - $f ( x ) = 4 x ^ { 4 } - 8 x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1$

Give the equation of the horizontal asymptote, if any, of the function. - $h ( x ) = \frac { 7 x ^ { 2 } - 5 x - 4 } { 8 x ^ { 2 } - 8 x + 9 }$

Use the Factor Theorem to determine whether x - c is a factor of f. If it is, write f in factored form, that is, write f in the form f(x) = (x - c)(quotient). - $f ( x ) = 5 x ^ { 4 } - 7 x ^ { 3 } + 17 x ^ { 2 } - 21 x + 6 ; c = 1$

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. - $x ^ { 4 } - x ^ { 3 } - 7 x ^ { 2 } + 5 x + 10 = 0 ; 2 < r \leq 3$

Find the x- and y-intercepts of f. - $f ( x ) = 5 x - x ^ { 3 }$

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $f ( x ) = - 15 x ^ { 4 } + \pi x ^ { 3 } + 1$

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

Review

Filters

Exam 3: Polynomial and Rational Functions

Find the vertical asymptotes of the rational function. - f(x)=−2x(x+2)4x2−5x−9f ( x ) = \frac { - 2 x ( x + 2 ) } { 4 x ^ { 2 } - 5 x - 9 }f(x)=4x2−5x−9−2x(x+2)​

Solve the equation in the real number system. - x3+5x2+2x−8=0x ^ { 3 } + 5 x ^ { 2 } + 2 x - 8 = 0x3+5x2+2x−8=0

Find the indicated intercept(s) of the graph of the function.2133:2139 - x-intercepts of f(x)=5x2−x−12x \text {-intercepts of } f ( x ) = \frac { 5 } { x ^ { 2 } - x - 12 }x-intercepts of f(x)=x2−x−125​

Use the given zero to find the remaining zeros of the function. - f(x)=x4−45x2−196; zero: −2if ( x ) = x ^ { 4 } - 45 x ^ { 2 } - 196 ; \text { zero: } - 2 if(x)=x4−45x2−196; zero: −2i

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. - x3−8x−3=0;−1≤r≤0\mathrm { x } ^ { 3 } - 8 \mathrm { x } - 3 = 0 ; - 1 \leq \mathrm { r } \leq 0x3−8x−3=0;−1≤r≤0

Give the equation of the horizontal asymptote, if any, of the function. - f(x)=x2−749x−x4f ( x ) = \frac { x ^ { 2 } - 7 } { 49 x - x ^ { 4 } }f(x)=49x−x4x2−7​

Solve the problem. - x2−5x≤0x ^ { 2 } - 5 x \leq 0x2−5x≤0

Solve the problem. - x3≥125x ^ { 3 } \geq 125x3≥125

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

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)=65−15xf ( x ) = \frac { 6 } { 5 } - \frac { 1 } { 5 } xf(x)=56​−51​x

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)=3(x2+5)(x+6)2f ( x ) = 3 \left( x ^ { 2 } + 5 \right) ( x + 6 ) ^ { 2 }f(x)=3(x2+5)(x+6)2

Find the x- and y-intercepts of f. - f(x)=−x2(x+5)(x2−1)f ( x ) = - x ^ { 2 } ( x + 5 ) \left( x ^ { 2 } - 1 \right)f(x)=−x2(x+5)(x2−1)

Graph the function using transformations. - f(x)=−2x+1f ( x ) = \frac { - 2 } { x + 1 }f(x)=x+1−2​ A) B) C) D)

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)=4x4−8x3+5x2−2x+1f ( x ) = 4 x ^ { 4 } - 8 x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1f(x)=4x4−8x3+5x2−2x+1

Graph the function using transformations. - f(x)=6−1(x+3)2f ( x ) = 6 - \frac { 1 } { ( x + 3 ) ^ { 2 } }f(x)=6−(x+3)21​ A) B) C) D)

Give the equation of the horizontal asymptote, if any, of the function. - h(x)=7x2−5x−48x2−8x+9h ( x ) = \frac { 7 x ^ { 2 } - 5 x - 4 } { 8 x ^ { 2 } - 8 x + 9 }h(x)=8x2−8x+97x2−5x−4​

Use the Factor Theorem to determine whether x - c is a factor of f. If it is, write f in factored form, that is, write f in the form f(x) = (x - c)(quotient). - f(x)=5x4−7x3+17x2−21x+6;c=1f ( x ) = 5 x ^ { 4 } - 7 x ^ { 3 } + 17 x ^ { 2 } - 21 x + 6 ; c = 1f(x)=5x4−7x3+17x2−21x+6;c=1

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. - x4−x3−7x2+5x+10=0;2<r≤3x ^ { 4 } - x ^ { 3 } - 7 x ^ { 2 } + 5 x + 10 = 0 ; 2 < r \leq 3x4−x3−7x2+5x+10=0;2<r≤3

Find the x- and y-intercepts of f. - f(x)=5x−x3f ( x ) = 5 x - x ^ { 3 }f(x)=5x−x3

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)=−15x4+πx3+1f ( x ) = - 15 x ^ { 4 } + \pi x ^ { 3 } + 1f(x)=−15x4+πx3+1