Question 1

Solve the equation in the real number system.
-$x ^ { 4 } - 21 x ^ { 2 } - 100 = 0$

Accepted Answer

A)  $\{ - 5,5 \}$ 
B)  $\{ - 5 , - 2,2,5 \}$ 
C)  $\{ - 10,10 \}$ 
D)  $\{ - 2,2 \}$ 
A)  $\{ - 5,5 \}$ 
B)  $\{ - 5 , - 2,2,5 \}$ 
C)  $\{ - 10,10 \}$ 
D)  $\{ - 2,2 \}$

Question 2

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

Accepted Answer

A)  none 
B)  $x = - 2 , x = 2$ 
C)  $x = - 2 , x = - 9$ 
D)  $x = - 2 , x = 2 , x = - 9$ 
A)  none 
B)  $x = - 2 , x = 2$ 
C)  $x = - 2 , x = - 9$ 
D)  $x = - 2 , x = 2 , x = - 9$

Question 3

Form a polynomial whose zeros and degree are given. Use a leading coefficient of 1.
-Zeros: 3, multiplicity 2; -3, multiplicity 2; degree 4 A) $f ( x ) = x ^ { 4 } - 6 x ^ { 3 } + 18 x ^ { 2 } - 27 x + 81$

Accepted Answer

B)  $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } - 18 x ^ { 2 } + 27 x - 81$ 
C)  $f ( x ) = x ^ { 4 } + 18 x ^ { 2 } + 81$ 
D)  $f ( x ) = x ^ { 4 } - 18 x ^ { 2 } + 81$ 
B)  $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } - 18 x ^ { 2 } + 27 x - 81$ 
C)  $f ( x ) = x ^ { 4 } + 18 x ^ { 2 } + 81$ 
D)  $f ( x ) = x ^ { 4 } - 18 x ^ { 2 } + 81$

Question 4

Find the indicated intercept(s) of the graph of the function.2133:2139
-$y$-intercept of $f ( x ) = \frac { ( 5 x - 15 ) ( x - 4 ) } { x ^ { 2 } + 12 x - 19 }$

Accepted Answer

A)  $\left( 0 , \frac { 60 } { 19 } \right)$ 
B)  $( 0,3 )$ 
C)  $\left( 0 , - \frac { 60 } { 19 } \right)$ 
D)  $( 0,5 )$ 
A)  $\left( 0 , \frac { 60 } { 19 } \right)$ 
B)  $( 0,3 )$ 
C)  $\left( 0 , - \frac { 60 } { 19 } \right)$ 
D)  $( 0,5 )$

Question 5

Find the vertical asymptotes of the rational function.
-$$h ( x ) = \frac { x + 4 } { x ^ { 2 } - 36 }$$

Accepted Answer

A)  $x = 36 , x = - 4$ 
B)  $x = - 6 , x = 6$ 
C)  $x = 0 , x = 36$ 
D)  $x = - 6 , x = 6 , x = - 4$ 
A)  $x = 36 , x = - 4$ 
B)  $x = - 6 , x = 6$ 
C)  $x = 0 , x = 36$ 
D)  $x = - 6 , x = 6 , x = - 4$

Question 6

Find the indicated intercept(s) of the graph of the function.2133:2139
-$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 6 x + 8 ; x - 4$

Accepted Answer

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

Question 7

Solve the problem.
-$$\mathrm { x } ^ { 2 } - 64 > 0$$

Accepted Answer

A)  $( - 8,8 )$ 
B)  $( - 64,64 )$ 
C)  $( - \infty , - 8 )$ or $( 8 , \infty )$ 
D)  $( - \infty , - 64 )$ or $( 64 , \infty )$ 
A)  $( - 8,8 )$ 
B)  $( - 64,64 )$ 
C)  $( - \infty , - 8 )$ or $( 8 , \infty )$ 
D)  $( - \infty , - 64 )$ or $( 64 , \infty )$

Question 8

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).
-$$f ( x ) = 2 x ^ { 3 } + 16 x ^ { 2 } + 39 x + 45 ; c = - 5$$

Accepted Answer

A)  No 
B)  Yes; $f ( x ) = ( x - 5 ) \left( 2 x ^ { 2 } + 6 x - 9 \right)$ 
C)  Yes; $f ( x ) = ( x + 5 ) \left( 2 x ^ { 2 } + x + 9 \right)$ 
D)  Yes; $f ( x ) = ( x + 5 ) \left( 2 x ^ { 2 } + 6 x + 9 \right)$ 
A)  No 
B)  Yes; $f ( x ) = ( x - 5 ) \left( 2 x ^ { 2 } + 6 x - 9 \right)$ 
C)  Yes; $f ( x ) = ( x + 5 ) \left( 2 x ^ { 2 } + x + 9 \right)$ 
D)  Yes; $f ( x ) = ( x + 5 ) \left( 2 x ^ { 2 } + 6 x + 9 \right)$

Question 9

Solve the inequality.
-$$\frac { ( x - 3 ) ( x + 3 ) } { x } \leq 0$$

Accepted Answer

A)  $[ - 3,0 )$ or $( 0,3 ]$ 
B)  $[ - 3,0 )$ or $[ 3 , \infty )$ 
C)  $( - \infty , - 3 ]$ or $( 0,3 ]$ 
D)  $( - \infty , - 3 ]$ or $[ 3 , \infty )$ 
A)  $[ - 3,0 )$ or $( 0,3 ]$ 
B)  $[ - 3,0 )$ or $[ 3 , \infty )$ 
C)  $( - \infty , - 3 ]$ or $( 0,3 ]$ 
D)  $( - \infty , - 3 ]$ or $[ 3 , \infty )$

Question 10

Give the equation of the oblique asymptote, if any, of the function.
-$$h ( x ) = \frac { 3 x ^ { 2 } - 8 x - 4 } { 9 x ^ { 2 } - 4 x + 6 }$$

Accepted Answer

A)  $y = x + \frac { 1 } { 3 }$ 
B)  $y = \frac { 1 } { 3 }$ 
C)  $y = \frac { 1 } { 3 } x$ 
D)  no oblique asymptote 
A)  $y = x + \frac { 1 } { 3 }$ 
B)  $y = \frac { 1 } { 3 }$ 
C)  $y = \frac { 1 } { 3 } x$ 
D)  no oblique asymptote

Question 11

Find the x- and y-intercepts of f.
-$$f ( x ) = ( x - 4 ) ( x - 1 )$$

Accepted Answer

A)  $x$-intercepts: 4,1 ; y-intercept: $- 5$ 
B)  $x$-intercepts: 4,1 ; y-intercept: 4 
C)  $x$-intercepts: $- 4 , - 1 ; y$-intercept: $- 5$ 
D)  x-intercepts: $- 4 , - 1 ; \mathrm { y }$-intercept: 4 
A)  $x$-intercepts: 4,1 ; y-intercept: $- 5$ 
B)  $x$-intercepts: 4,1 ; y-intercept: 4 
C)  $x$-intercepts: $- 4 , - 1 ; y$-intercept: $- 5$ 
D)  x-intercepts: $- 4 , - 1 ; \mathrm { y }$-intercept: 4

Question 12

Give the equation of the oblique asymptote, if any, of the function.
-$$f ( x ) = \frac { x ^ { 2 } - 6 } { 36 x - x ^ { 4 } }$$

Accepted Answer

A)  y = 36x 
B)  y = 0 
C)  no oblique asymptote 
D)  y = x - 6 
A)  y = 36x 
B)  y = 0 
C)  no oblique asymptote 
D)  y = x - 6

Question 13

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

Accepted Answer

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

Question 14

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at
each x -intercept.
-$$f ( x ) = \frac { 1 } { 5 } x \left( x ^ { 2 } - 5 \right)$$

Accepted Answer

A)  0 , multiplicity 1 
B)  $\sqrt { 5 }$, multiplicity 1 , touches $x$-axis;
$- \sqrt { 5 }$, multiplicity 1 , touches $x$-axis 
C)  0 , multiplicity 1 , crosses x-axis;
$\sqrt { 5 }$, multiplicity 1 , touches $x$-axis;
$- \sqrt { 5 }$, multiplicity 1 , crosses $x$-axis 
D)  0 , multiplicity 1 , touches $x$-axis;
$\sqrt { 5 }$, multiplicity 1 , crosses $x$-axis;
$- \sqrt { 5 }$, multiplicity 1 , touches $x$-axis 
A)  0 , multiplicity 1 
B)  $\sqrt { 5 }$, multiplicity 1 , touches $x$-axis;
$- \sqrt { 5 }$, multiplicity 1 , touches $x$-axis 
C)  0 , multiplicity 1 , crosses x-axis;
$\sqrt { 5 }$, multiplicity 1 , touches $x$-axis;
$- \sqrt { 5 }$, multiplicity 1 , crosses $x$-axis 
D)  0 , multiplicity 1 , touches $x$-axis;
$\sqrt { 5 }$, multiplicity 1 , crosses $x$-axis;
$- \sqrt { 5 }$, multiplicity 1 , touches $x$-axis

Question 15

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

Question 16

Solve the problem.
-A can in the shape of a right circular cylinder is required to have a volume of 700 cubic centimeters. The top and bottom are made up of a material that costs 8¢ per square centimeter, while the sides are made of material that
Costs 5¢ per square centimeter. Which function below describes the total cost of the material as a function of the
Radius r of the cylinder? A) $C ( r ) = 0.16 \pi r ^ { 2 } + \frac { 140 } { r }$

Accepted Answer

B)  $C ( r ) = 0.08 \pi r ^ { 2 } + \frac { 70 } { r }$ 
C)  $C ( r ) = 0.16 \pi r ^ { 2 } + \frac { 70 } { r }$ 
D)  $C ( r ) = 0.08 \pi r ^ { 2 } + \frac { 140 } { r }$ 
B)  $C ( r ) = 0.08 \pi r ^ { 2 } + \frac { 70 } { r }$ 
C)  $C ( r ) = 0.16 \pi r ^ { 2 } + \frac { 70 } { r }$ 
D)  $C ( r ) = 0.08 \pi r ^ { 2 } + \frac { 140 } { r }$

Question 17

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 18

Solve the problem.
-$$2 x ^ { 2 } + 5 x < 3$$

Accepted Answer

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

Question 19

Form a polynomial f(x) with real coefficients having the given degree and zeros.
-Degree: 3 ; zeros: $- 3$ and $3 - 2 \mathrm { i }$

Accepted Answer

A)  $f ( x ) = x ^ { 3 } - x ^ { 2 } + 11 x + 39$ 
B)  $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 5 x - 52$ 
C)  $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39$ 
D)  $f ( x ) = x ^ { 3 } - x ^ { 2 } - 5 x + 39$ 
A)  $f ( x ) = x ^ { 3 } - x ^ { 2 } + 11 x + 39$ 
B)  $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 5 x - 52$ 
C)  $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39$ 
D)  $f ( x ) = x ^ { 3 } - x ^ { 2 } - 5 x + 39$

Question 20

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)  $f ( x ) = ( x - i \sqrt { 6 } ) ( x + i \sqrt { 6 } ) ( x - 3 ) ( x + 3 )$

Solve the equation in the real number system. - $x ^ { 4 } - 21 x ^ { 2 } - 100 = 0$

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

Find the indicated intercept(s) of the graph of the function.2133:2139 - $y$ -intercept of $f ( x ) = \frac { ( 5 x - 15 ) ( x - 4 ) } { x ^ { 2 } + 12 x - 19 }$

Find the vertical asymptotes of the rational function. - $h ( x ) = \frac { x + 4 } { x ^ { 2 } - 36 }$

Find the indicated intercept(s) of the graph of the function.2133:2139 - $f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 6 x + 8 ; x - 4$

Solve the problem. - $\mathrm { x } ^ { 2 } - 64 > 0$

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). - $f ( x ) = 2 x ^ { 3 } + 16 x ^ { 2 } + 39 x + 45 ; c = - 5$

Solve the inequality. - $\frac { ( x - 3 ) ( x + 3 ) } { x } \leq 0$

Give the equation of the oblique asymptote, if any, of the function. - $h ( x ) = \frac { 3 x ^ { 2 } - 8 x - 4 } { 9 x ^ { 2 } - 4 x + 6 }$

Find the x- and y-intercepts of f. - $f ( x ) = ( x - 4 ) ( x - 1 )$

Give the equation of the oblique asymptote, if any, of the function. - $f ( x ) = \frac { x ^ { 2 } - 6 } { 36 x - x ^ { 4 } }$

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. - $f ( x ) = \frac { 1 } { 5 } x \left( x ^ { 2 } - 5 \right)$

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. - $f ( x ) = x ^ { 6 } - x ^ { 5 } - x ^ { 4 } - 5 x ^ { 3 } + 2 x ^ { 2 } + 2 x + 3$

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

Solve the problem. - $2 x ^ { 2 } + 5 x < 3$

Form a polynomial f(x) with real coefficients having the given degree and zeros. -Degree: 3 ; zeros: $- 3$ and $3 - 2 \mathrm { i }$

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

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

Solve the equation in the real number system. - x4−21x2−100=0x ^ { 4 } - 21 x ^ { 2 } - 100 = 0x4−21x2−100=0

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

Find the indicated intercept(s) of the graph of the function.2133:2139 - yyy -intercept of f(x)=(5x−15)(x−4)x2+12x−19f ( x ) = \frac { ( 5 x - 15 ) ( x - 4 ) } { x ^ { 2 } + 12 x - 19 }f(x)=x2+12x−19(5x−15)(x−4)​

Find the vertical asymptotes of the rational function. - h(x)=x+4x2−36h ( x ) = \frac { x + 4 } { x ^ { 2 } - 36 }h(x)=x2−36x+4​

Find the indicated intercept(s) of the graph of the function.2133:2139 - f(x)=x3+2x2−6x+8;x−4f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 6 x + 8 ; x - 4f(x)=x3+2x2−6x+8;x−4

Solve the problem. - x2−64>0\mathrm { x } ^ { 2 } - 64 > 0x2−64>0

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)=2x3+16x2+39x+45;c=−5f ( x ) = 2 x ^ { 3 } + 16 x ^ { 2 } + 39 x + 45 ; c = - 5f(x)=2x3+16x2+39x+45;c=−5

Solve the inequality. - (x−3)(x+3)x≤0\frac { ( x - 3 ) ( x + 3 ) } { x } \leq 0x(x−3)(x+3)​≤0

Give the equation of the oblique asymptote, if any, of the function. - h(x)=3x2−8x−49x2−4x+6h ( x ) = \frac { 3 x ^ { 2 } - 8 x - 4 } { 9 x ^ { 2 } - 4 x + 6 }h(x)=9x2−4x+63x2−8x−4​

Find the x- and y-intercepts of f. - f(x)=(x−4)(x−1)f ( x ) = ( x - 4 ) ( x - 1 )f(x)=(x−4)(x−1)

Give the equation of the oblique asymptote, if any, of the function. - f(x)=x2−636x−x4f ( x ) = \frac { x ^ { 2 } - 6 } { 36 x - x ^ { 4 } }f(x)=36x−x4x2−6​

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)=15x(x2−5)f ( x ) = \frac { 1 } { 5 } x \left( x ^ { 2 } - 5 \right)f(x)=51​x(x2−5)

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​

Solve the problem. - 2x2+5x<32 x ^ { 2 } + 5 x < 32x2+5x<3

Form a polynomial f(x) with real coefficients having the given degree and zeros. -Degree: 3 ; zeros: −3- 3−3 and 3−2i3 - 2 \mathrm { i }3−2i

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