Question 1

Solve the equation in the real number system.
-$$2 x ^ { 4 } - 2 x ^ { 3 } + x ^ { 2 } - 5 x - 10 = 0$$

Accepted Answer

A)  $\left\{ - \frac { 5 } { 2 } , \frac { 5 } { 2 } \right\}$ 
B)  $\left\{ - \frac { \sqrt { 10 } } { 2 } , \frac { \sqrt { 10 } } { 2 } \right\}$ 
C)  $\{ 1 , - 2 \}$ 
D)  $\{ - 1,2 \}$ 
A)  $\left\{ - \frac { 5 } { 2 } , \frac { 5 } { 2 } \right\}$ 
B)  $\left\{ - \frac { \sqrt { 10 } } { 2 } , \frac { \sqrt { 10 } } { 2 } \right\}$ 
C)  $\{ 1 , - 2 \}$ 
D)  $\{ - 1,2 \}$

Question 2

Write the word or phrase that best completes each statement or answers the question.
-A box has a base whose length is twice its width. The volume of the box is 7000 cubic inches.
a) Find a function for the surface area of the box.
b) What are the dimensions of the box that minimizes surface area?

Accepted Answer

a) Surface area equa

Question 3

Find the x- and y-intercepts of f.
-$$f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } - 1 \right)$$

Accepted Answer

A)  $x$-intercepts: $- 6 , - 1,0,1 ; y$-intercept: $- 6$ 
B)  x-intercepts: $- 6,0,1 ;$ y-intercept: $- 6$ 
C)  $x$-intercepts: $- 6 , - 1,0,1 ; \mathrm { y }$-intercept: 0 
D)  x-intercepts: $- 1,0,1,6 ; y$-intercept: 0 
A)  $x$-intercepts: $- 6 , - 1,0,1 ; y$-intercept: $- 6$ 
B)  x-intercepts: $- 6,0,1 ;$ y-intercept: $- 6$ 
C)  $x$-intercepts: $- 6 , - 1,0,1 ; \mathrm { y }$-intercept: 0 
D)  x-intercepts: $- 1,0,1,6 ; y$-intercept: 0

Question 4

Find the domain of the rational function.
-$$g ( x ) = \frac { x + 7 } { x ^ { 2 } + 1 }$$

Accepted Answer

A)  all real numbers 
B)  $\{ x \mid x \neq 0 , x \neq - 1 \}$ 
C)  $\{ x \mid x \neq - 1 , x \neq 1 \}$ 
D)  $\{ x \mid x \neq - 1 , x \neq 1 , x \neq - 7 \}$ 
A)  all real numbers 
B)  $\{ x \mid x \neq 0 , x \neq - 1 \}$ 
C)  $\{ x \mid x \neq - 1 , x \neq 1 \}$ 
D)  $\{ x \mid x \neq - 1 , x \neq 1 , x \neq - 7 \}$

Question 5

Use the graph to find the vertical asymptotes, if any, of the function.
-

Accepted Answer

A)  $x = 0$ 
B)  $x = 0 , y = 0$ 
C)  none 
D)  $y = - 8 , y = 8$ 
A)  $x = 0$ 
B)  $x = 0 , y = 0$ 
C)  none 
D)  $y = - 8 , y = 8$

Question 6

Find the power function that the graph of f resembles for large values of |x|.
-$$f ( x ) = ( x + 8 ) ^ { 2 }$$

Accepted Answer

A)  $y = x ^ { 16 }$ 
B)  $y = x ^ { 2 }$ 
C)  $y = x ^ { 8 }$ 
D)  $y = x ^ { 64 }$ 
A)  $y = x ^ { 16 }$ 
B)  $y = x ^ { 2 }$ 
C)  $y = x ^ { 8 }$ 
D)  $y = x ^ { 64 }$

Question 7

Use the graph to find the horizontal asymptote, if any, of the function.
-

Accepted Answer

A)  y = -4, y = 4 
B)  x = -4, x = 4, y = 1 
C)  y = 1 
D)  y = 0, y = 1 
A)  y = -4, y = 4 
B)  x = -4, x = 4, y = 1 
C)  y = 1 
D)  y = 0, y = 1

Question 8

Use the x-intercepts to find the intervals on which the graph of f is above and below the x-axis.
-$$f ( x ) = \left( x + \frac { 1 } { 8 } \right) ^ { 2 } ( x - 2 ) ^ { 5 }$$

Accepted Answer

A)  above the $x$-axis: $\left( - \frac { 1 } { 8 } , 2 \right)$
below the X-axis: $\left( - \infty , - \frac { 1 } { 8 } \right) , ( 2 , \infty )$ 
B)  above the $x$-axis: $\left( - \infty , - \frac { 1 } { 8 } \right) , ( 2 , \infty )$
below the x-axis: $\left( - \frac { 1 } { 8 } , 2 \right)$ 
C)  above the $x$-axis: $\left( - \infty , - \frac { 1 } { 8 } \right) , \left( - \frac { 1 } { 8 } , 2 \right)$
below the $x$-axis: $( 2 , \infty )$ 
D)  above the $x$-axis: $( 2 , \infty )$
below the $x$-axis: $\left( - \infty , - \frac { 1 } { 8 } \right) , \left( - \frac { 1 } { 8 } , 2 \right)$ 
A)  above the $x$-axis: $\left( - \frac { 1 } { 8 } , 2 \right)$
below the X-axis: $\left( - \infty , - \frac { 1 } { 8 } \right) , ( 2 , \infty )$ 
B)  above the $x$-axis: $\left( - \infty , - \frac { 1 } { 8 } \right) , ( 2 , \infty )$
below the x-axis: $\left( - \frac { 1 } { 8 } , 2 \right)$ 
C)  above the $x$-axis: $\left( - \infty , - \frac { 1 } { 8 } \right) , \left( - \frac { 1 } { 8 } , 2 \right)$
below the $x$-axis: $( 2 , \infty )$ 
D)  above the $x$-axis: $( 2 , \infty )$
below the $x$-axis: $\left( - \infty , - \frac { 1 } { 8 } \right) , \left( - \frac { 1 } { 8 } , 2 \right)$

Question 9

Solve the inequality.
-$$( x - 2 ) ( x - 3 ) > 0$$

Accepted Answer

A)  $( 2,3 )$ 
B)  $( - \infty , 2 )$ 
C)  $( 3 , \infty )$ 
D)  $( - \infty , 2 )$ or $( 3 , \infty )$ 
A)  $( 2,3 )$ 
B)  $( - \infty , 2 )$ 
C)  $( 3 , \infty )$ 
D)  $( - \infty , 2 )$ or $( 3 , \infty )$

Question 10

Choose the one alternative that best completes the statement or answers the question.
Find a bound on the real zeros of the polynomial function.
-$$f ( x ) = 6 x ^ { 3 } - x ^ { 2 } + 0.3 x - 0.06$$

Accepted Answer

A)  $- 6$ and 6 
B)  $- 1$ and 1 
C)  $- 1.36$ and $1.36$ 
D)  $- 2$ and 2 
A)  $- 6$ and 6 
B)  $- 1$ and 1 
C)  $- 1.36$ and $1.36$ 
D)  $- 2$ and 2

Question 11

Choose the one alternative that best completes the statement or answers the question.
-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 e per square centimeter. Which function below describes the total cost of the material as a function of the radius $\mathrm { r }$ of the cylinder?

Accepted Answer

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

Question 12

List the potential rational zeros of the polynomial function. Do not find the zeros.
-$$f ( x ) = 7 x ^ { 3 } - x ^ { 2 } + 2$$

Accepted Answer

A)  $\pm \frac { 1 } { 7 } , \pm \frac { 2 } { 7 } , \pm 1 , \pm 2$ 
B)  $\pm \frac { 1 } { 7 } , \pm \frac { 1 } { 2 } , \pm 1 , \pm 2 , \pm 7$ 
C)  $\pm \frac { 1 } { 7 } , \pm \frac { 2 } { 7 } , \pm 1 , \pm 2 , \pm 7$ 
D)  $\pm \frac { 1 } { 2 } , \pm \frac { 7 } { 2 } , \pm 1 , \pm 7$ 
A)  $\pm \frac { 1 } { 7 } , \pm \frac { 2 } { 7 } , \pm 1 , \pm 2$ 
B)  $\pm \frac { 1 } { 7 } , \pm \frac { 1 } { 2 } , \pm 1 , \pm 2 , \pm 7$ 
C)  $\pm \frac { 1 } { 7 } , \pm \frac { 2 } { 7 } , \pm 1 , \pm 2 , \pm 7$ 
D)  $\pm \frac { 1 } { 2 } , \pm \frac { 7 } { 2 } , \pm 1 , \pm 7$

Question 13

Use the graph to find the vertical asymptotes, if any, of the function.
-

Accepted Answer

A)  $x = - 4 , x = 4 , x = 0 , y = 1$ 
B)  $x = - 4 , x = 4 , x = 0$ 
C)  $x = - 4 , x = 4$ 
D)  $x = - 4 , x = 4 , y = 1$ 
A)  $x = - 4 , x = 4 , x = 0 , y = 1$ 
B)  $x = - 4 , x = 4 , x = 0$ 
C)  $x = - 4 , x = 4$ 
D)  $x = - 4 , x = 4 , y = 1$

Question 14

Use the Factor Theorem to determine whether x - c is a factor of f(x).
-$f ( x ) = 56 x ^ { 3 } + 61 x ^ { 2 } - 102 x - 110 ; x + \frac { 11 } { 8 }$

Accepted Answer

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

Question 15

Find the indicated intercept(s) of the graph of the function.
-$y$-intercept of $f ( x ) = \frac { ( x - 6 ) ^ { 2 } } { ( x + 11 ) ^ { 3 } }$

Accepted Answer

A)  $\left( 0 , - \frac { 6 } { 11 } \right)$ 
B)  $\left( 0 , \frac { 36 } { 1331 } \right)$ 
C)  $( 0,6 )$ 
D)  $\left( 0 , - \frac { 36 } { 1331 } \right)$ 
A)  $\left( 0 , - \frac { 6 } { 11 } \right)$ 
B)  $\left( 0 , \frac { 36 } { 1331 } \right)$ 
C)  $( 0,6 )$ 
D)  $\left( 0 , - \frac { 36 } { 1331 } \right)$

Question 16

Give the equation of the horizontal asymptote, if any, of the function.
-$$f ( x ) = \frac { 3 x ^ { 2 } + 4 } { 3 x ^ { 2 } - 4 }$$

Accepted Answer

A)  $y = 4$ 
B)  $y = 3$ 
C)  $y = 1$ 
D)  no horizontal asymptotes 
A)  $y = 4$ 
B)  $y = 3$ 
C)  $y = 1$ 
D)  no horizontal asymptotes

Question 17

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 18

Choose the one alternative that best completes the statement or answers the question.
-A closed box with a square base has to have a volume of 16,000 cubic inches. Find a function for the surface area of the box.

Accepted Answer

A)  $S ( x ) = x ^ { 2 } + \frac { 64,000 } { x }$ 
B)  $S ( x ) = 2 x ^ { 2 } + \frac { 64,000 } { x }$ 
C)  $S ( x ) = 2 x ^ { 2 } + \frac { 16,000 } { x }$ 
D)  $S ( x ) = 2 x ^ { 2 } + \frac { 96,000 } { x }$ 
A)  $S ( x ) = x ^ { 2 } + \frac { 64,000 } { x }$ 
B)  $S ( x ) = 2 x ^ { 2 } + \frac { 64,000 } { x }$ 
C)  $S ( x ) = 2 x ^ { 2 } + \frac { 16,000 } { x }$ 
D)  $S ( x ) = 2 x ^ { 2 } + \frac { 96,000 } { x }$

Question 19

Solve the problem.
-The polynomial function $\mathrm { f } ( \mathrm { x } ) = 6 \mathrm { x } ^ { 3 } + 19 \mathrm { x } ^ { 2 } + 8 \mathrm { x } - 5$ has exactly one positive zero.
Use the Intermediate Value Theorem to approximate the zero correct to 2 decimal places.

Accepted Answer

A)  $0.10$ 
B)  $0.66$ 
C)  $0.33$ 
D)  $0.50$ 
A)  $0.10$ 
B)  $0.66$ 
C)  $0.33$ 
D)  $0.50$

Question 20

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.
-$$f ( x ) = x ^ { 4 } - 5 x ^ { 2 } - 36$$

Accepted Answer

A)  $3 ; f ( x ) = ( x - 3 ) ^ { 2 } \left( x ^ { 2 } + 4 \right)$ 
B)  $- 3,3 ; f ( x ) = ( x - 3 ) ( x + 3 ) \left( x ^ { 2 } + 4 \right)$ 
C)  $- 3 , - 2,3,2 ; f ( x ) = ( x - 3 ) ( x + 3 ) ( x - 2 ) ( x + 2 )$ 
D)  $- 2,2 ; f ( x ) = ( x - 2 ) ( x + 2 ) \left( x ^ { 2 } + 9 \right)$ 
A)  $3 ; f ( x ) = ( x - 3 ) ^ { 2 } \left( x ^ { 2 } + 4 \right)$ 
B)  $- 3,3 ; f ( x ) = ( x - 3 ) ( x + 3 ) \left( x ^ { 2 } + 4 \right)$ 
C)  $- 3 , - 2,3,2 ; f ( x ) = ( x - 3 ) ( x + 3 ) ( x - 2 ) ( x + 2 )$ 
D)  $- 2,2 ; f ( x ) = ( x - 2 ) ( x + 2 ) \left( x ^ { 2 } + 9 \right)$

Solve the equation in the real number system. - $2 x ^ { 4 } - 2 x ^ { 3 } + x ^ { 2 } - 5 x - 10 = 0$

Write the word or phrase that best completes each statement or answers the question. -A box has a base whose length is twice its width. The volume of the box is 7000 cubic inches. a) Find a function for the surface area of the box. b) What are the dimensions of the box that minimizes surface area?

Find the x- and y-intercepts of f. - $f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } - 1 \right)$

Find the domain of the rational function. - $g ( x ) = \frac { x + 7 } { x ^ { 2 } + 1 }$

Use the graph to find the vertical asymptotes, if any, of the function. -

Find the power function that the graph of f resembles for large values of |x|. - $f ( x ) = ( x + 8 ) ^ { 2 }$

Use the graph to find the horizontal asymptote, if any, of the function. -

Use the x-intercepts to find the intervals on which the graph of f is above and below the x-axis. - $f ( x ) = \left( x + \frac { 1 } { 8 } \right) ^ { 2 } ( x - 2 ) ^ { 5 }$

Solve the inequality. - $( x - 2 ) ( x - 3 ) > 0$

Choose the one alternative that best completes the statement or answers the question. Find a bound on the real zeros of the polynomial function. - $f ( x ) = 6 x ^ { 3 } - x ^ { 2 } + 0.3 x - 0.06$

List the potential rational zeros of the polynomial function. Do not find the zeros. - $f ( x ) = 7 x ^ { 3 } - x ^ { 2 } + 2$

Use the graph to find the vertical asymptotes, if any, of the function. -

Use the Factor Theorem to determine whether x - c is a factor of f(x). - $f ( x ) = 56 x ^ { 3 } + 61 x ^ { 2 } - 102 x - 110 ; x + \frac { 11 } { 8 }$

Find the indicated intercept(s) of the graph of the function. - $y$ -intercept of $f ( x ) = \frac { ( x - 6 ) ^ { 2 } } { ( x + 11 ) ^ { 3 } }$

Give the equation of the horizontal asymptote, if any, of the function. - $f ( x ) = \frac { 3 x ^ { 2 } + 4 } { 3 x ^ { 2 } - 4 }$

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

Choose the one alternative that best completes the statement or answers the question. -A closed box with a square base has to have a volume of 16,000 cubic inches. Find a function for the surface area of the box.

Solve the problem. -The polynomial function $\mathrm { f } ( \mathrm { x } ) = 6 \mathrm { x } ^ { 3 } + 19 \mathrm { x } ^ { 2 } + 8 \mathrm { x } - 5$ has exactly one positive zero. Use the Intermediate Value Theorem to approximate the zero correct to 2 decimal places.

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. - $f ( x ) = x ^ { 4 } - 5 x ^ { 2 } - 36$

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

Solve the equation in the real number system. - 2x4−2x3+x2−5x−10=02 x ^ { 4 } - 2 x ^ { 3 } + x ^ { 2 } - 5 x - 10 = 02x4−2x3+x2−5x−10=0

Write the word or phrase that best completes each statement or answers the question. -A box has a base whose length is twice its width. The volume of the box is 7000 cubic inches. a) Find a function for the surface area of the box. b) What are the dimensions of the box that minimizes surface area?

Find the x- and y-intercepts of f. - f(x)=−x2(x+6)(x2−1)f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } - 1 \right)f(x)=−x2(x+6)(x2−1)

Find the domain of the rational function. - g(x)=x+7x2+1g ( x ) = \frac { x + 7 } { x ^ { 2 } + 1 }g(x)=x2+1x+7​

Use the graph to find the vertical asymptotes, if any, of the function. -

Find the power function that the graph of f resembles for large values of |x|. - f(x)=(x+8)2f ( x ) = ( x + 8 ) ^ { 2 }f(x)=(x+8)2

Use the graph to find the horizontal asymptote, if any, of the function. -

Use the x-intercepts to find the intervals on which the graph of f is above and below the x-axis. - f(x)=(x+18)2(x−2)5f ( x ) = \left( x + \frac { 1 } { 8 } \right) ^ { 2 } ( x - 2 ) ^ { 5 }f(x)=(x+81​)2(x−2)5

Solve the inequality. - (x−2)(x−3)>0( x - 2 ) ( x - 3 ) > 0(x−2)(x−3)>0

Choose the one alternative that best completes the statement or answers the question. Find a bound on the real zeros of the polynomial function. - f(x)=6x3−x2+0.3x−0.06f ( x ) = 6 x ^ { 3 } - x ^ { 2 } + 0.3 x - 0.06f(x)=6x3−x2+0.3x−0.06

List the potential rational zeros of the polynomial function. Do not find the zeros. - f(x)=7x3−x2+2f ( x ) = 7 x ^ { 3 } - x ^ { 2 } + 2f(x)=7x3−x2+2

Use the graph to find the vertical asymptotes, if any, of the function. -

Use the Factor Theorem to determine whether x - c is a factor of f(x). - f(x)=56x3+61x2−102x−110;x+118f ( x ) = 56 x ^ { 3 } + 61 x ^ { 2 } - 102 x - 110 ; x + \frac { 11 } { 8 }f(x)=56x3+61x2−102x−110;x+811​

Find the indicated intercept(s) of the graph of the function. - yyy -intercept of f(x)=(x−6)2(x+11)3f ( x ) = \frac { ( x - 6 ) ^ { 2 } } { ( x + 11 ) ^ { 3 } }f(x)=(x+11)3(x−6)2​

Give the equation of the horizontal asymptote, if any, of the function. - f(x)=3x2+43x2−4f ( x ) = \frac { 3 x ^ { 2 } + 4 } { 3 x ^ { 2 } - 4 }f(x)=3x2−43x2+4​

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

Choose the one alternative that best completes the statement or answers the question. -A closed box with a square base has to have a volume of 16,000 cubic inches. Find a function for the surface area of the box.

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)=x4−5x2−36f ( x ) = x ^ { 4 } - 5 x ^ { 2 } - 36f(x)=x4−5x2−36

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

Solve the equation in the real number system. - $2 x ^ { 4 } - 2 x ^ { 3 } + x ^ { 2 } - 5 x - 10 = 0$

Find the x- and y-intercepts of f. - $f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } - 1 \right)$

Find the domain of the rational function. - $g ( x ) = \frac { x + 7 } { x ^ { 2 } + 1 }$

Find the power function that the graph of f resembles for large values of |x|. - $f ( x ) = ( x + 8 ) ^ { 2 }$

Use the x-intercepts to find the intervals on which the graph of f is above and below the x-axis. - $f ( x ) = \left( x + \frac { 1 } { 8 } \right) ^ { 2 } ( x - 2 ) ^ { 5 }$

Solve the inequality. - $( x - 2 ) ( x - 3 ) > 0$

Choose the one alternative that best completes the statement or answers the question. Find a bound on the real zeros of the polynomial function. - $f ( x ) = 6 x ^ { 3 } - x ^ { 2 } + 0.3 x - 0.06$

List the potential rational zeros of the polynomial function. Do not find the zeros. - $f ( x ) = 7 x ^ { 3 } - x ^ { 2 } + 2$

Use the Factor Theorem to determine whether x - c is a factor of f(x). - $f ( x ) = 56 x ^ { 3 } + 61 x ^ { 2 } - 102 x - 110 ; x + \frac { 11 } { 8 }$

Find the indicated intercept(s) of the graph of the function. - $y$ -intercept of $f ( x ) = \frac { ( x - 6 ) ^ { 2 } } { ( x + 11 ) ^ { 3 } }$

Give the equation of the horizontal asymptote, if any, of the function. - $f ( x ) = \frac { 3 x ^ { 2 } + 4 } { 3 x ^ { 2 } - 4 }$

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

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. - $f ( x ) = x ^ { 4 } - 5 x ^ { 2 } - 36$