Question 1

State the domain of the rational function.
-$$f ( x ) = \frac { x ^ { 2 } + 4 x - 12 } { x ^ { 2 } - 2 x - 8 }$$
A) $( - \infty , - 4 ) \cup ( - 4,2 ) \cup ( 2 , \infty )$
B) $( - \infty , - 2 ) \cup ( - 2,4 ) \cup ( 4 , \infty )$
C) $( - \infty , \infty )$
D) $( - \infty , 4 ) \cup ( 4 , \infty )$

Accepted Answer

The answer of State the domain of the rational function.
-\[f...

Question 2

Solve.
-An open-top rectangular box has a square base and it will hold 256 cubic centimeters (cc). Each side of the base has length $x \mathrm {~cm}$. The box's surface area $S$ is given by
$$S ( x ) = \frac { 1024 } { x } + x ^ { 2 } .$$
Estimate the minimum surface area and the value of $x$ that will yield it.
A) $256 \mathrm {~cm} ^ { 2 }$ when $x = 6 \mathrm {~cm}$
B) $207 \mathrm {~cm} ^ { 2 }$ when $x = 6 \mathrm {~cm}$
C) $256 \mathrm {~cm} ^ { 2 }$ when $x = 8 \mathrm {~cm}$
D) $192 \mathrm {~cm} ^ { 2 }$ when $\mathrm { x } = 8 \mathrm {~cm}$

Accepted Answer

To find the minimum surface area, we differentiate $S(x)$ with respect to $x$ and set the derivative equal to zero to find critical points. The derivative of $S(x)$ is $S'(x) = -1024/x^2 + 2x$. Setting $S'(x) = 0$ gives $-1024/x^2 + 2x = 0$, which simplifies to $2x^3 = 1024$, leading to $x^3 = 512$, and thus $x = 8$. Substituting $x = 8$ into the original equation for $S(x)$ gives $S(8) = 1024/8 + 8^2 = 128 + 64 = 192 \mathrm{cm}^2$.

Question 3

Find only the rational zeros.
-$$f ( x ) = x ^ { 5 } - 5 x ^ { 4 } + 5 x ^ { 3 } + 15 x ^ { 2 } - 36 x + 20$$
A) $- 1,2 , - 2$
B) $1,2,3$
C) $1,2 , - 2$
D) No rational zeros

Accepted Answer

The rational zeros of a polynomial can be found using the Rational Root Theorem, which states that any rational zero, expressed as a fraction $\frac{p}{q}$, must have $p$ as a factor of the constant term and $q$ as a factor of the leading coefficient. For the given polynomial, the constant term is 20 and the leading coefficient is 1. The factors of 20 are $\pm1, \pm2, \pm4, \pm5, \pm10, \pm20$, and since the leading coefficient is 1, the possible rational zeros are these factors themselves. Testing these values or using synthetic division, we find that $1, 2, \text{ and } -2$ are indeed zeros of the polynomial, making option C correct.

Question 4

Find the horizontal asymptote, if any, of the rational function.
-$$f ( x ) = \frac { 5 } { x ^ { 2 } + 2 }$$
A) $y = 5$
B) $y = 0$
C) $x = 0$
D) None

Accepted Answer

The horizontal asymptote of a rational function $f(x) = \frac{p(x)}{q(x)}$ is found by comparing the degrees of the polynomials $p(x)$ and $q(x)$. If the degree of $q(x)$ is greater than the degree of $p(x)$, as in this case (degree of $q(x) = 2$, degree of $p(x) = 0$), the horizontal asymptote is $y = 0$.

Question 5

Provide the requested response.
-Suppose that a polynomial function of degree 5 with rational coefficients has $- 5,4 , - 3,5 - i$ as zeros. Find the other zero(s).
A) $5 + i$
B) $- 5 - \mathrm { i }$
C) $5 , - 4,3,5 + \mathrm { i }$
D) $- 5 + \mathrm { i }$

Accepted Answer

If a polynomial has rational coefficients, then any complex zeros must come in conjugate pairs. Since $5 - i$ is a zero, its conjugate $5 + i$ must also be a zero.

Question 6

Use long division to determine whether the binomial is a factor of f(x).
-$$f ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 31 x + 70 ; x + 5$$
A) Yes
B) $\mathrm { No }$

Accepted Answer

When dividing $f(x) = x^3 - 4x^2 - 31x + 70$ by $x + 5$, the remainder is 0, indicating that $x + 5$ is a factor of $f(x)$.

Question 7

For the function find the maximum number of real zeros that the function can have, the maximum number of x-intercepts
that the function can have, and the maximum number of turning points that the graph of the function can have.
-$$f ( x ) = - x ^ { 6 } + 8 x ^ { 5 } - x ^ { 2 } - 4 x + 9$$
A) $6 ; 6 ; 5$
B) $6 ; 5 ; 5$
C) $6 ; 6 ; 6$
D) $5 ; 5 ; 6$

Accepted Answer

The maximum number of real zeros a polynomial can have is equal to its degree, which is 6 for this polynomial. The maximum number of x-intercepts on a graph is equal to the maximum number of real zeros, so it's also 6. The maximum number of turning points for a polynomial of degree n is n-1, which is 5 for a 6th degree polynomial.

Question 8

List the critical values of the related function. Then solve the inequality.
-$$\frac { 2 } { x ^ { 2 } + 1 } \geq \frac { 4 } { 4 x ^ { 2 } + 3 }$$
A) No critical values; $( - \infty , \infty )$
B) $2 ; ( - \infty , 2 ]$
C) No critical values; $\varnothing$
D) $4 ; [ 4 , \infty )$

Accepted Answer

To solve the inequality $\frac { 2 } { x ^ { 2 } + 1 } \geq \frac { 4 } { 4 x ^ { 2 } + 3 }$, we first find a common denominator and simplify the inequality. Both denominators are always positive for all real x, so there are no critical values from the denominators turning zero. Solving the inequality does not lead to a specific interval but shows the original inequality is always true for all x, hence the solution is $( - \infty , \infty )$.

Question 9

Factor the polynomial f(x). Then solve the equation f(x) = 0.
-$$f ( x ) = x ^ { 3 } + 11 x ^ { 2 } + 36 x + 36$$
A) $( x + 2 ) ( x + 3 ) ( x + 6 ) ; 2,3,6$
B) $( x + 2 ) ( x + 3 ) ( x + 6 ) ; - 2 , - 3 , - 6$
C) $( x - 3 ) ( x - 4 ) ( x - 7 ) ; 3,4,7$
D) $( x + 3 ) ( x + 4 ) ( x + 7 ) ; 3,4,7$

Accepted Answer

To factor the polynomial $f(x) = x^3 + 11x^2 + 36x + 36$, we look for factors of 36 that add up to 11 for the middle term. The correct factorization is $(x + 2)(x + 3)(x + 6)$, which gives the roots $-2, -3, -6$.

Question 10

Graph the function, showing all asymptotes (those that do not correspond to an axis) as dashed lines. List the x- and
y-intercepts.
-$$f ( x ) = \frac { x ^ { 2 } - 3 x - 4 } { 3 x ^ { 2 } + 2 }$$

A) No $x$-intercepts, $y$-intercept: $( 0 , - 4 )$;

B) No $x$-intercepts, $y$-intercept: $( 0 , - 4 )$;

C) x-intercepts: $( 1,0 )$ and $( - 4,0 )$,

D) x-intercepts: $( - 1,0 )$ and $( 4,0 )$, y-intercept: $( 0 , - 2 )$; $\mathrm { y }$-intercept: $( 0 , - 2 )$;

Accepted Answer

The answer of Graph the function, showing all asymptotes (those...

Question 11

List the critical values of the related function. Then solve the inequality.
-$$\frac { 1 } { x - 4 } \leq 0$$
A) $- 4,4 ; [ - 4,4 ]$
B) $4 ; ( - \infty , 4 )$
C) No critical values; $\varnothing$
D) $- 4 ; ( - \infty , - 4 ]$

Accepted Answer

The critical value of the function is $x = 4$, because the inequality involves a fraction with a denominator that can become 0, making the function undefined at $x = 4$. The inequality $\frac{1}{x-4} \leq 0$ is satisfied for $x < 4$, hence the solution is $x \in (-\infty, 4)$.

Question 12

Graph the function.
-$$h ( x ) = x ^ { 5 } - 17 x ^ { 2 } + 16 x$$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the function.
-\[h ( x ) =...

Question 13

Provide the requested response.
-Suppose that a polynomial function of degree 4 with rational coefficients has 6, 4, 3i as zeros. Find the other zero.
A)3- i
B)- 3i
C)-6 $$\text { D) } \sqrt { 3 } \mathrm { i }$$

Accepted Answer

Since the polynomial has rational coefficients, the non-real zeros must come in conjugate pairs. Given 3i as a zero, its conjugate -3i must also be a zero.

Question 14

Solve.
-The average cost per tape, in dollars, for a company to produce x sports videotapes is given by the function $$A ( x ) = \frac { 14 x + 80 } { x } \text { for } x > 0$$
Graph the function on the interval $( 0 , \infty )$ and complete the following:
$$\mathrm { A } ( \mathrm { x } ) \rightarrowــــــــــ \quad \text { as } \mathrm { x } \rightarrow \infty \text {. }$$ 
A)

$\mathrm { A } ( \mathrm { x } ) \rightarrow 1$ as $\mathrm { x } \rightarrow \infty$.
B)

$\mathrm { A } ( \mathrm { x } ) \rightarrow 0$ as $\mathrm { x } \rightarrow \infty .$
C)

$\mathrm { A } ( \mathrm { x } ) \rightarrow 19$ as $\mathrm { x } \rightarrow \infty .$
D)

$\mathrm { A } ( \mathrm { x } ) \rightarrow 14$ as $\mathrm { x } \rightarrow \infty$.

Accepted Answer

The answer of Solve.
-The average cost per tape, in dollars,...

Question 15

Find a rational function that satisfies the given conditions. Answers may vary, but try to give the simplest answer
possible.
-Oblique asymptote $y = x + 5$
A) $f ( x ) = \frac { x ^ { 2 } - 5 x + 1 } { 5 x }$
B) $f ( x ) = \frac { x ^ { 2 } + 5 x + 1 } { x }$
C) $f ( x ) = \frac { x ^ { 2 } + 5 x + 1 } { 5 x }$
D) $f ( x ) = \frac { x ^ { 2 } - 5 x + 1 } { x }$

Accepted Answer

An oblique (or slant) asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. The given oblique asymptote is $y = x + 5$, which means the division of the numerator by the denominator should result in a quotient of $x + 5$ plus a remainder. Option B, $f(x) = \frac{x^2 + 5x + 1}{x}$, directly matches this requirement because dividing $x^2 + 5x + 1$ by $x$ gives a quotient of $x + 5$ plus a remainder of $1/x$, which becomes negligible as $x$ approaches infinity, leaving the oblique asymptote $y = x + 5$.

Question 16

Classify the polynomial as constant, linear, quadratic, cubic, or quartic, and determine the leading term, the leading
coefficient, and the degree of the polynomial.
-$$g ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 12 x + 8$$
A) Linear; $- 12 x ; - 12 ; 1$
B) Quadratic; $\frac { 1 } { 2 } x ^ { 2 } ; \frac { 1 } { 2 } ; 2$
C) Cubic; $\frac { 1 } { 2 } x ^ { 3 } ; \frac { 1 } { 2 } ; 3$
D) Linear; $8 ; 8 ; 1$

Accepted Answer

The polynomial is cubic because the highest power of $x$ is 3. The leading term is $\frac{1}{2}x^3$, the leading coefficient is $\frac{1}{2}$, and the degree is 3.

Question 17

Given that the polynomial function has the given zero, find the other zeros.
-$$f ( x ) = x ^ { 4 } - 21 x ^ { 2 } - 100 ; - 2 i$$
A) $2 \mathrm { i } , 5 \mathrm { i } , - 5 \mathrm { i }$
B) $2 \mathrm { i } , 10 , - 10$
C) $2 \mathrm { i } , 5 , - 5$
D) $2 \mathrm { i } , 10 \mathrm { i } , - 10 \mathrm { i }$

Accepted Answer

The answer of Given that the polynomial function has the...

Question 18

Given that the polynomial function has the given zero, find the other zeros.
-$$f ( x ) = x ^ { 3 } - 64 ; 4$$
A) $- 2 + \sqrt { 3 } i , - 2 - \sqrt { 3 } i$
B) $- 2 + 2 \sqrt { 3 } i , - 2 - 2 \sqrt { 3 } i$
C) $- 1 + 2 \sqrt { 3 } i , - 1 - 2 \sqrt { 3 } i$
D) $- 2 + 4 \sqrt { 3 } i , - 2 - 4 \sqrt { 3 } i$

Accepted Answer

The answer of Given that the polynomial function has the...

Question 19

Solve the inequality.
-For the function $f ( x ) = x ^ { 2 } - 2 x - 35$, solve $f ( x ) \leq 0$.
A) $( - \infty , - 5 ] \cup [ 7 , \infty )$
B) $[ - 5,7 ]$
C) $( - \infty , - 5 ]$
D) $[ 7 , \infty )$

Accepted Answer

To solve the inequality $f(x) \leq 0$, first find the roots of the quadratic equation $x^2 - 2x - 35 = 0$. Factoring the equation gives $(x - 7)(x + 5) = 0$, so the roots are $x = 7$ and $x = -5$. The inequality $f(x) \leq 0$ is satisfied between these roots, so the solution is $[ - 5,7 ]$.

Question 20

Use synthetic division to find the function value.
-$f ( x ) = x ^ { 5 } - 10 x ^ { 4 } + 15 x ^ { 3 } - 4 x - 300 ;$ find $f ( 2 )$
A) $- 314$
B) 284
C) $- 318$
D) $- 316$

Accepted Answer

The answer of Use synthetic division to find the function...

Quiz 4: Polynomial and Rational Functions

State the domain of the rational function. - $f ( x ) = \frac { x ^ { 2 } + 4 x - 12 } { x ^ { 2 } - 2 x - 8 }$

Find only the rational zeros. - $f ( x ) = x ^ { 5 } - 5 x ^ { 4 } + 5 x ^ { 3 } + 15 x ^ { 2 } - 36 x + 20$

Find the horizontal asymptote, if any, of the rational function. - $f ( x ) = \frac { 5 } { x ^ { 2 } + 2 }$

Provide the requested response. -Suppose that a polynomial function of degree 5 with rational coefficients has $- 5,4 , - 3,5 - i$ as zeros. Find the other zero(s) .

Use long division to determine whether the binomial is a factor of f(x) . - $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 31 x + 70 ; x + 5$

For the function find the maximum number of real zeros that the function can have, the maximum number of x-intercepts that the function can have, and the maximum number of turning points that the graph of the function can have. - $f ( x ) = - x ^ { 6 } + 8 x ^ { 5 } - x ^ { 2 } - 4 x + 9$

List the critical values of the related function. Then solve the inequality. - $\frac { 2 } { x ^ { 2 } + 1 } \geq \frac { 4 } { 4 x ^ { 2 } + 3 }$

Factor the polynomial f(x) . Then solve the equation f(x) = 0. - $f ( x ) = x ^ { 3 } + 11 x ^ { 2 } + 36 x + 36$

List the critical values of the related function. Then solve the inequality. - $\frac { 1 } { x - 4 } \leq 0$

Graph the function. - $h ( x ) = x ^ { 5 } - 17 x ^ { 2 } + 16 x$ $Graph the function. - h ( x ) = x ^ { 5 } - 17 x ^ { 2 } + 16 x A) B) C) D)$

Provide the requested response. -Suppose that a polynomial function of degree 4 with rational coefficients has 6, 4, 3i as zeros. Find the other zero.

Find a rational function that satisfies the given conditions. Answers may vary, but try to give the simplest answer possible. -Oblique asymptote $y = x + 5$

Classify the polynomial as constant, linear, quadratic, cubic, or quartic, and determine the leading term, the leading coefficient, and the degree of the polynomial. - $g ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 12 x + 8$

Given that the polynomial function has the given zero, find the other zeros. - $f ( x ) = x ^ { 4 } - 21 x ^ { 2 } - 100 ; - 2 i$

Given that the polynomial function has the given zero, find the other zeros. - $f ( x ) = x ^ { 3 } - 64 ; 4$

Solve the inequality. -For the function $f ( x ) = x ^ { 2 } - 2 x - 35$ , solve $f ( x ) \leq 0$ .

Use synthetic division to find the function value. - $f ( x ) = x ^ { 5 } - 10 x ^ { 4 } + 15 x ^ { 3 } - 4 x - 300 ;$ find $f ( 2 )$

Quiz 4: Polynomial and Rational Functions

State the domain of the rational function. - f(x) =x2+4x−12x2−2x−8f ( x ) = \frac { x ^ { 2 } + 4 x - 12 } { x ^ { 2 } - 2 x - 8 }f(x) =x2−2x−8x2+4x−12​

Find only the rational zeros. - f(x) =x5−5x4+5x3+15x2−36x+20f ( x ) = x ^ { 5 } - 5 x ^ { 4 } + 5 x ^ { 3 } + 15 x ^ { 2 } - 36 x + 20f(x) =x5−5x4+5x3+15x2−36x+20

Find the horizontal asymptote, if any, of the rational function. - f(x) =5x2+2f ( x ) = \frac { 5 } { x ^ { 2 } + 2 }f(x) =x2+25​

Provide the requested response. -Suppose that a polynomial function of degree 5 with rational coefficients has −5,4,−3,5−i- 5,4 , - 3,5 - i−5,4,−3,5−i as zeros. Find the other zero(s) .

Use long division to determine whether the binomial is a factor of f(x) . - f(x) =x3−4x2−31x+70;x+5f ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 31 x + 70 ; x + 5f(x) =x3−4x2−31x+70;x+5

List the critical values of the related function. Then solve the inequality. - 2x2+1≥44x2+3\frac { 2 } { x ^ { 2 } + 1 } \geq \frac { 4 } { 4 x ^ { 2 } + 3 }x2+12​≥4x2+34​

Factor the polynomial f(x) . Then solve the equation f(x) = 0. - f(x) =x3+11x2+36x+36f ( x ) = x ^ { 3 } + 11 x ^ { 2 } + 36 x + 36f(x) =x3+11x2+36x+36

Graph the function, showing all asymptotes (those that do not correspond to an axis) as dashed lines. List the x- and y-intercepts. - f(x) =x2−3x−43x2+2f ( x ) = \frac { x ^ { 2 } - 3 x - 4 } { 3 x ^ { 2 } + 2 }f(x) =3x2+2x2−3x−4​

List the critical values of the related function. Then solve the inequality. - 1x−4≤0\frac { 1 } { x - 4 } \leq 0x−41​≤0

Graph the function. - h(x) =x5−17x2+16xh ( x ) = x ^ { 5 } - 17 x ^ { 2 } + 16 xh(x) =x5−17x2+16x