Content check — College Algebra, Chapter 5: Polynomial and Rational Functions

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 5: Polynomial and Rational Functions

381 problems · 75 figures

Try It #1 §5.1 source ↗

A coordinate grid has been superimposed over the quadratic path of a basketball in Figure 8. Assume that the point (–4, 7) is the highest point of the basketball’s trajectory. Find an equation for the path of the ball. Does the shooter make the basket?

Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes.

Figure 8 (credit: modification of work by Dan Meyer)

Answer

The path passes through the origin and has vertex at

(- 4, 7),

h (x) =

- \frac{7}{16} {(x + 4)}^{2} +

7.

To make the shot,

h (- 7.5)

would need to be about 4 but

h (- 7.5) \approx

1.64;

he doesn’t make it.

Try It #2 §5.1 source ↗

Given the equation

g (x) =

13 +

x^{2} -

6 x,

write the equation in general form and then in standard form.

Answer

g (x) =

x^{2} -

6 x +

13

in general form;

g (x) =

{(x - 3)}^{2} +

4

in standard form

Try It #3 §5.1 source ↗

Find the domain and range of

f (x) =

2 {(x - \frac{4}{7})}^{2} +

\frac{8}{11} .

Answer

The domain is all real numbers. The range is

f (x) \geq \frac{8}{11},

[\frac{8}{11}, \infty) .

Try It #4 §5.1 source ↗

In a Try It, we found the standard and general form for the function

g (x) =

13 +

x^{2} -

6 x .

Now find the y- and x-intercepts (if any).

Answer

y-intercept at (0, 13), No

x -

intercepts

Try It #5 §5.1 source ↗

A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. The rock’s height above ocean can be modeled by the equation $H (t) =$ $−16 t^{2} +$ $96 t +$ $112.$

ⓐWhen does the rock reach the maximum height?

ⓑWhat is the maximum height of the rock?

ⓒWhen does the rock hit the ocean?

Answer

ⓐ3 seconds

ⓑ256 feet

ⓒ7 seconds

5.1-1 §5.1 source ↗

Explain the advantage of writing a quadratic function in standard form.

Answer

When written in that form, the vertex can be easily identified.

5.1-3 §5.1 source ↗

Explain why the condition of

a \neq 0

is imposed in the definition of the quadratic function.

Answer

a = 0

then the function becomes a linear function.

5.1-5 §5.1 source ↗

What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?

Answer

If possible, we can use factoring. Otherwise, we can use the quadratic formula.

5.1-7 §5.1 source ↗

For the following exercises, rewrite the quadratic functions in vertex form and give the vertex.

$g (x) =$ $x^{2} +$ $2 x -$ $3$

Answer

g (x) =

{(x + 1)}^{2} -

4,

Vertex

(- 1, - 4)

5.1-9 §5.1 source ↗

For the following exercises, rewrite the quadratic functions in vertex form and give the vertex.

$f (x) =$ $x^{2} +$ $5 x -$ $2$

Answer

f (x) =

{(x + \frac{5}{2})}^{2} -

\frac{33}{4},

Vertex

(- \frac{5}{2}, - \frac{33}{4})

5.1-11 §5.1 source ↗

For the following exercises, rewrite the quadratic functions in vertex form and give the vertex.

$k (x) =$ $3 x^{2} -$ $6 x -$ $9$

Answer

f (x) =

3 {(x - 1)}^{2} -

12,

Vertex

(1, - 12)

5.1-13 §5.1 source ↗

For the following exercises, rewrite the quadratic functions in vertex form and give the vertex.

$f (x) =$ $3 x^{2} -$ $5 x -$ $1$

Answer

f (x) =

3 {(x - \frac{5}{6})}^{2} -

\frac{37}{12},

Vertex

(\frac{5}{6}, - \frac{37}{12})

5.1-15 §5.1 source ↗

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.

$f (x) =$ $2 x^{2} -$ $10 x +$ $4$

Answer

Minimum is

- \frac{17}{2}

and occurs at

\frac{5}{2} .

Axis of symmetry is

x = \frac{5}{2} .

5.1-17 §5.1 source ↗

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.

$f (x) =$ $4 x^{2} +$ $x -$ $1$

Answer

Minimum is

- \frac{17}{16}

and occurs at

- \frac{1}{8} .

Axis of symmetry is

x = - \frac{1}{8} .

5.1-19 §5.1 source ↗

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.

$f (x) =$ $\frac{1}{2} x^{2} +$ $3 x +$ $1$

Answer

Minimum is

- \frac{7}{2}

and occurs at

−3.

Axis of symmetry is

x = −3.

5.1-21 §5.1 source ↗

For the following exercises, determine the domain and range of the quadratic function.

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

Answer

Domain is

(- \infty, \infty) .

Range is

[2, \infty) .

5.1-23 §5.1 source ↗

For the following exercises, determine the domain and range of the quadratic function.

$f (x) =$ $x^{2} +$ $6 x +$ $4$

Answer

Domain is

(- \infty, \infty) .

Range is

[−5, \infty) .

5.1-25 §5.1 source ↗

For the following exercises, determine the domain and range of the quadratic function.

$k (x) =$ $3 x^{2} -$ $6 x -$ $9$

Answer

Domain is

(- \infty, \infty) .

Range is

[−12, \infty) .

5.1-27 §5.1 source ↗

For the following exercises, use the vertex $(h, k)$ and a point on the graph $(x, y)$ to find the general form of the equation of the quadratic function.

$(h, k) =$ $(−2, −1),$ $(x, y) =$ $(−4, 3)$

Answer

f (x) =

x^{2} +

4 x +

3

5.1-29 §5.1 source ↗

For the following exercises, use the vertex $(h, k)$ and a point on the graph $(x, y)$ to find the general form of the equation of the quadratic function.

$(h, k) =$ $(2, 3),$ $(x, y) =$ $(5, 12)$

Answer

f (x) =

x^{2} - 4 x +

7

5.1-31 §5.1 source ↗

For the following exercises, use the vertex $(h, k)$ and a point on the graph $(x, y)$ to find the general form of the equation of the quadratic function.

$(h, k) =$ $(3, 2),$ $(x, y) =$ $(10, 1)$

Answer

f (x) =

- \frac{1}{49} x^{2} +

\frac{6}{49} x +

\frac{89}{49}

5.1-33 §5.1 source ↗

For the following exercises, use the vertex $(h, k)$ and a point on the graph $(x, y)$ to find the general form of the equation of the quadratic function.

$(h, k) =$ $(1, 0),$ $(x, y) =$ $(0, 1)$

Answer

f (x) =

x^{2} - 2 x +

1

5.1-35 §5.1 source ↗

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.

$f (x) =$ $x^{2} -$ $6 x -$ $1$

Answer

Vertex: (3, −10), axis of symmetry: x = 3, intercepts: $(3 + \sqrt{10}, 0)$ and $(3 - \sqrt{10}, 0)$

A parabola on a Cartesian coordinate system. The parabola opens upwards, has its vertex in the fourth quadrant, and intersects the x-axis at (0,0) and approximately (7,0).

5.1-37 §5.1 source ↗

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.

$f (x) =$ $x^{2} -$ $7 x +$ $3$

Answer

Vertex: $(\frac{7}{2}, - \frac{37}{4})$ , axis of symmetry: $x = \frac{7}{2}$ , y-intercept: $(0, 3)$ , x-intercepts: $(\frac{7 + \sqrt{37}}{2}, 0),$ $(\frac{7 - \sqrt{37}}{2}, 0)$

5.1-39 §5.1 source ↗

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.

$f (x) =$ $4 x^{2} -$ $12 x -$ $3$

Answer

Vertex: $(\frac{3}{2}, - 12)$ , axis of symmetry: $x = \frac{3}{2}$ , intercept: $(\frac{3 + 2 \sqrt{3}}{2}, 0)$ and $(\frac{3 - 2 \sqrt{3}}{2}, 0)$

A graph displays a blue parabola opening upwards. The parabola's vertex is in the fourth quadrant, intersecting the x-axis around (3.5, 0) and the y-axis around (0, -3).

5.1-41 §5.1 source ↗

For the following exercises, write the equation for the graphed quadratic function.

Graph of a positive parabola with a vertex at (-1, 2) and y-intercept at (0, 3)

Answer

f (x) =

x^{2} +

2 x +

3

5.1-43 §5.1 source ↗

For the following exercises, write the equation for the graphed quadratic function.

Graph of a negative parabola with a vertex at (-1, 2).

Answer

f (x) =

- 3 x^{2} -

6 x -

1

5.1-45 §5.1 source ↗

For the following exercises, write the equation for the graphed quadratic function.

Graph of a negative parabola with a vertex at (-2, 3).

Answer

f (x) =

- \frac{1}{4} x^{2} -

x +

2

5.1-47 §5.1 source ↗

For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

$x$	–2	–1	0	1	2
$y$	1	0	1	4	9

Answer

f (x) =

x^{2} +

2 x +

1

5.1-49 §5.1 source ↗

$x$	–2	–1	0	1	2
$y$	–8	–3	0	1	0

Answer

f (x) = - x^{2} + 2 x

5.1-50 §5.1 source ↗

$x$	–2	–1	0	1	2
$y$	8	2	0	2	8

Answer

f (x) = 2 x^{2}

5.1-53 §5.1 source ↗

For the following exercises, use a calculator to find the answer.

Graph on the same set of axes $f (x) =$ $x^{2},$ $f (x) =$ ${(x - 2)}^{2},$ $f {(x - 3)}^{2}$ , and $f (x) =$ ${(x + 4)}^{2} .$

What appears to be the effect of adding or subtracting those numbers?

Answer

The graph is shifted to the right or left (a horizontal shift).

5.1-55 §5.1 source ↗

For the following exercises, use a calculator to find the answer.

A suspension bridge can be modeled by the quadratic function $h (x) =$ $.0001 x^{2}$ with $−2000 \leq x \leq 2000$ where $| x |$ is the number of feet from the center and $h (x)$ is height in feet. Use the TRACE feature of your calculator to estimate how far from the center does the bridge have a height of 100 feet.

Answer

The suspension bridge has 1,000 feet distance from the center.

5.1-57 §5.1 source ↗

For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.

Vertex $(−1, 2)$ opens down.

Answer

Domain is

(- \infty, \infty) .

Range is

(- \infty, 2] .

5.1-59 §5.1 source ↗

For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.

Vertex $(−100, 100),$ opens up.

Answer

Domain:

(- \infty, \infty)

; range:

[100, \infty)

5.1-61 §5.1 source ↗

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Contains $(−1, 4)$ and has the shape of $f (x) = 2 x^{2} .$ Vertex is on the $y -$ axis.

Answer

f (x) = 2 x^{2} + 2

5.1-63 §5.1 source ↗

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Contains $(1, −3)$ and has the shape of $f (x) = - x^{2} .$ Vertex is on the $y -$ axis.

Answer

f (x) = - x^{2} - 2

5.1-65 §5.1 source ↗

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Contains $(1, −6)$ has the shape of $f (x) = 3 x^{2} .$ Vertex has x-coordinate of $−1.$

Answer

f (x) =

3 x^{2} +

6 x - 15

5.1-67 §5.1 source ↗

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Find the dimensions of the rectangular dog park split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing.

Answer

75 feet by 50 feet

5.1-69 §5.1 source ↗

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product?

Answer

3 and 3; product is 9

5.1-71 §5.1 source ↗

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Suppose that the price per unit in dollars of a cell phone production is modeled by $p =$ $$ 45 -$ $0.0125 x,$ where $x$ is in thousands of phones produced, and the revenue represented by thousands of dollars is $R = x \cdot p .$ Find the production level that will maximize revenue.

Answer

The revenue reaches the maximum value when 1800 thousand phones are produced.

5.1-73 §5.1 source ↗

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by $h (t) =$ $-$ $4.9 t^{2} +$ $24 t +$ $8.$ How long does it take to reach maximum height?

Answer

2.449 seconds

5.1-75 §5.1 source ↗

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

Answer

41 trees per acre

Try It #1 §5.2 source ↗

Which functions are power functions?

$\begin{matrix} f (x) & = & 2 x \cdot 4 x^{3} \\ g (x) & = & - x^{5} + 5 x^{3} \\ h (x) & = & \frac{2 x^{5} - 1}{3 x^{2} + 4} \end{matrix}$

Answer

f (x)

is a power function because it can be written as

f (x) = 8 x^{4} .

The other functions are not power functions.

Try It #2 §5.2 source ↗

Describe in words and symbols the end behavior of

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

Answer

x

approaches positive or negative infinity,

f (x)

decreases without bound: as

x \to

\pm \infty,

f (x) \to

-

\infty

because of the negative coefficient.

Try It #3 §5.2 source ↗

Identify the degree, leading term, and leading coefficient of the polynomial

f (x) =

4 x^{2} -

x^{6} +

2 x -

6.

Answer

The degree is 6. The leading term is

- x^{6} .

The leading coefficient is

- 1.

Try It #4 §5.2 source ↗

Describe the end behavior, and determine a possible degree of the polynomial function in Figure 8.

Figure 8

Answer

x \to

\infty,

f (x) \to

-

\infty;

a s x \to

-

\infty,

f (x) \to

-

\infty .

It has the shape of an even degree power function with a negative coefficient.

Try It #5 §5.2 source ↗

Given the function

f (x) =

0.2 (x - 2) (x + 1) (x - 5),

express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.

Answer

The leading term is

0.2 x^{3},

so it is a degree 3 polynomial. As

x

approaches positive infinity,

f (x)

increases without bound; as

x

approaches negative infinity,

f (x)

decreases without bound.

Try It #6 §5.2 source ↗

Given the polynomial function

f (x) =

2 x^{3} -

6 x^{2} -

20 x,

determine the y- and x-intercepts.

Answer

y-intercept

(0, 0);

x-intercepts

(0, 0),

(- 2, 0),

and

(5, 0)

Try It #7 §5.2 source ↗

Without graphing the function, determine the maximum number of x-intercepts and turning points for

f (x) =

108 -

13 x^{9} -

8 x^{4} +

14 x^{12} +

2 x^{3} .

Answer

There are at most 12

x -

intercepts and at most 11 turning points.

Try It #8 §5.2 source ↗

What can we conclude about the polynomial represented by the graph shown in Figure 14 based on its intercepts and turning points?

Figure 14

Answer

The end behavior indicates an odd-degree polynomial function; there are 3

x -

intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.

Try It #9 §5.2 source ↗

Given the function

f (x) =

0.2 (x - 2) (x + 1) (x - 5),

determine the local behavior.

Answer

The

x -

intercepts are

(2, 0),

(- 1, 0),

and

(5, 0),

the y-intercept is

(0, 2),

and the graph has at most 2 turning points.

5.2-1 §5.2 source ↗

Explain the difference between the coefficient of a power function and its degree.

Answer

The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

5.2-3 §5.2 source ↗

In general, explain the end behavior of a polynomial with odd degree if the leading coefficient is positive.

Answer

x

decreases without bound, so does

f (x) .

x

increases without bound, so does

f (x) .

5.2-5 §5.2 source ↗

What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As

x \to

-

\infty,

f (x) \to

-

\infty

and as

x \to

\infty,

f (x) \to

-

\infty .

Answer

The polynomial function is of even degree and leading coefficient is negative.

5.2-7 §5.2 source ↗

For the following exercises, identify the function as a power function, a polynomial function, or neither.

$f (x) = {(x^{2})}^{3}$

Answer

Power function

5.2-9 §5.2 source ↗

For the following exercises, identify the function as a power function, a polynomial function, or neither.

$f (x) = \frac{x^{2}}{x^{2} - 1}$

Answer

Neither

5.2-11 §5.2 source ↗

For the following exercises, identify the function as a power function, a polynomial function, or neither.

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

Answer

Neither

5.2-13 §5.2 source ↗

For the following exercises, find the degree and leading coefficient for the given polynomial.

$7 - 2 x^{2}$

Answer

Degree = 2, Coefficient = –2

5.2-15 §5.2 source ↗

For the following exercises, find the degree and leading coefficient for the given polynomial.

$x (4 - x^{2}) (2 x + 1)$

Answer

Degree =4, Coefficient = –2

5.2-17 §5.2 source ↗

For the following exercises, determine the end behavior of the functions.

$f (x) = x^{4}$

Answer

x \to \infty

f (x) \to

\infty,

as x \to

-

\infty,

f (x) \to

\infty

5.2-19 §5.2 source ↗

For the following exercises, determine the end behavior of the functions.

$f (x) = - x^{4}$

Answer

x \to - \infty

f (x) \to

-

\infty,

as x \to

\infty,

f (x) \to

-

\infty

5.2-21 §5.2 source ↗

For the following exercises, determine the end behavior of the functions.

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

Answer

x \to - \infty

f (x) \to

-

\infty,

as x \to

\infty,

f (x) \to

-

\infty

5.2-23 §5.2 source ↗

For the following exercises, determine the end behavior of the functions.

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

Answer

x \to \infty

f (x) \to

\infty,

as x \to

-

\infty,

f (x) \to

-

\infty

5.2-25 §5.2 source ↗

For the following exercises, find the intercepts of the functions.

$f (t) =$ $2 (t - 1) (t + 2) (t - 3)$

Answer

y-intercept is

(0, 12),

t-intercepts are

(1, 0);

(- 2, 0);

and (3, 0) .

5.2-27 §5.2 source ↗

For the following exercises, find the intercepts of the functions.

$f (x) = x^{4} - 16$

Answer

y-intercept is

(0, - 16) .

x-intercepts are

(2, 0)

and

(- 2, 0) .

5.2-29 §5.2 source ↗

For the following exercises, find the intercepts of the functions.

$f (x) =$ $x (x^{2} - 2 x - 8)$

Answer

y-intercept is

(0, 0) .

x-intercepts are

(0, 0),

(4, 0),

and

(- 2, 0) .

5.2-31 §5.2 source ↗

For the following exercises, determine the least possible degree of the polynomial function shown.

Answer

5.2-33 §5.2 source ↗

For the following exercises, determine the least possible degree of the polynomial function shown.

Answer

5.2-35 §5.2 source ↗

For the following exercises, determine the least possible degree of the polynomial function shown.

Answer

5.2-37 §5.2 source ↗

For the following exercises, determine the least possible degree of the polynomial function shown.

Answer

5.2-39 §5.2 source ↗

For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.

Answer

Yes. Number of turning points is 2. Least possible degree is 3.

5.2-41 §5.2 source ↗

Answer

Yes. Number of turning points is 1. Least possible degree is 2.

5.2-43 §5.2 source ↗

Answer

Yes. Number of turning points is 0. Least possible degree is 1.

5.2-45 §5.2 source ↗

Answer

Yes. Number of turning points is 0. Least possible degree is 1.

5.2-47 §5.2 source ↗

For the following exercises, make a table to confirm the end behavior of the function.

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

Answer

$x$	$f (x)$
10	9,500
100	99,950,000
–10	9,500
–100	99,950,000

As $x \to - \infty$ , $f (x) \to$ $\infty,$ $as x \to$ $\infty,$ $f (x) \to$ $\infty$

5.2-49 §5.2 source ↗

For the following exercises, make a table to confirm the end behavior of the function.

$f (x) =$ $(x - 1) (x - 2) (3 - x)$

Answer

$x$	$f (x)$
10	–504
100	–941,094
–10	1,716
–100	1,061,106

As $x \to - \infty$ , $f (x) \to$ $\infty,$ $as x \to$ $\infty,$ $f (x) \to$ $-$ $\infty$

5.2-51 §5.2 source ↗

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

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

Answer

The $y -$ intercept is $(0, 0) .$ The $x -$ intercepts are $(0, 0),$ $(2, 0) .$ As $x \to - \infty$ , $f (x) \to$ $\infty,$ $as x \to$ $\infty,$ $f (x) \to$ $\infty$

5.2-53 §5.2 source ↗

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

$f (x) =$ $x (14 - 2 x) (10 - 2 x)$

Answer

The $y -$ intercept is $(0, 0)$ . The $x -$ intercepts are $(0, 0),$ $(5, 0),$ $(7, 0) .$ As $x \to - \infty$ , $f (x) \to$ $-$ $\infty,$ $as x \to$ $\infty,$ $f (x) \to$ $\infty$

5.2-55 §5.2 source ↗

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

$f (x) = x^{3} - 16 x$

Answer

A graph on a coordinate plane displays a cubic function. The x-axis ranges from -10 to 10, and the y-axis ranges from -500 to 500. The blue curve illustrates the function, starting from approximately (-7, -400), increasing to a local maximum near (-1, 50), then decreasing to a local minimum near (3, -50), and finally increasing again, passing through (0, 0), (2, 0) and approximately (7, 300).

The $y -$ intercept is $(0, 0) .$ The $x -$ intercept is $(- 4, 0),$ $(0, 0),$ $(4, 0) .$ $A s x \to - \infty$ , $f (x) \to$ $-$ $\infty,$ $as x \to$ $\infty,$ $f (x) \to$ $\infty$

5.2-57 §5.2 source ↗

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

$f (x) = x^{4} - 81$

Answer

The $y -$ intercept is $(0, - 81) .$ The $x -$ intercept are $(3, 0),$ $(- 3, 0) .$ As $x \to - \infty$ , $f (x) \to$ $\infty,$ $as x \to$ $\infty,$ $f (x) \to$ $\infty$

5.2-59 §5.2 source ↗

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

$f (x) =$ $x^{3} -$ $2 x^{2} -$ $15 x$

Answer

The $y -$ intercept is $(0, 0) .$ The $x -$ intercepts are $(- 3, 0),$ $(0, 0),$ $(5, 0) .$ As $x \to - \infty$ , $f (x) \to$ $-$ $\infty,$ $as x \to$ $\infty,$ $f (x) \to$ $\infty$

5.2-61 §5.2 source ↗

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

The $y -$ intercept is $(0, - 4) .$ The $x -$ intercepts are $(- 2, 0)$ , $(2, 0) .$ Degree is 2.

End behavior: as $x \to - \infty$ , $f (x) \to \infty$ ; as $x \to \infty$ , $f (x) \to \infty .$

Answer

f (x) = x^{2} - 4

5.2-63 §5.2 source ↗

The $y -$ intercept is $(0, 0) .$ The $x -$ intercepts are $(0, 0)$ , $(2, 0) .$ Degree is 3.

End behavior: as $x \to - \infty$ , $f (x) \to - \infty$ , as $x \to \infty$ , $f (x) \to \infty .$

Answer

f (x) =

x^{3} -

4 x^{2} +

4 x

5.2-65 §5.2 source ↗

The $y -$ intercept is $(0, 1) .$ There is no $x -$ intercept. Degree is 4.

End behavior: as $x \to - \infty$ , $f (x) \to \infty$ , as $x \to \infty$ , $f (x) \to \infty .$

Answer

f (x) = x^{4} + 1

5.2-67 §5.2 source ↗

For the following exercises, use the written statements to construct a polynomial function that represents the required information.

A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of $m,$ the number of minutes elapsed.

Answer

V (m) =

8 m^{3} +

36 m^{2} +

54 m +

27

5.2-69 §5.2 source ↗

For the following exercises, use the written statements to construct a polynomial function that represents the required information.

An open box is to be constructed by cutting out square corners of $x -$ inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of $x .$

Answer

V (x) =

4 x^{3} -

32 x^{2} +

64 x

Try It #1 §5.3 source ↗

Find the y- and x-intercepts of the function

f (x) =

x^{4} -

19 x^{2} +

30 x .

Answer

y-intercept

(0, 0);

x-intercepts

(0, 0),

(- 5, 0),

(2, 0),

and

(3, 0)

Try It #2 §5.3 source ↗

Use the graph of the function of degree 9 in Figure 10 to identify the zeros of the function and their multiplicities.

Graph of an even-degree polynomial with degree 6.

Figure 10

Answer

The graph has a zero of –5 with multiplicity 3, a zero of -1 with multiplicity 2, and a zero of 3 with multiplicity 4.

Try It #3 §5.3 source ↗

Sketch a graph of

f (x) =

\frac{1}{4} x {(x - 1)}^{4} {(x + 3)}^{3} .

Answer

Try It #4 §5.3 source ↗

Show that the function

f (x) =

7 x^{5} -

9 x^{4} -

x^{2}

has at least one real zero between

x = 1

and

x = 2.

Answer

Because

f

is a polynomial function and since

f (1)

is negative and

f (2)

is positive, there is at least one real zero between

x = 1

and

x = 2.

Try It #5 §5.3 source ↗

Given the graph shown in Figure 20, write a formula for the function shown.

Graph of a negative even-degree polynomial with zeros at x=-1, 2, 4 and y=-4.

Figure 20

Answer

f (x) =

-

\frac{1}{8} {(x - 2)}^{3} {(x + 1)}^{2} (x - 4)

Try It #6 §5.3 source ↗

Use technology to find the maximum and minimum values on the interval

[−1, 4]

of the function

f (x) =

-

0.2 {(x - 2)}^{3} {(x + 1)}^{2} (x - 4) .

Answer

The minimum occurs at approximately the point

(0, - 6.5),

and the maximum occurs at approximately the point

(3.5, 7) .

5.3-1 §5.3 source ↗

What is the difference between an

x -

intercept and a zero of a polynomial function

f ?

Answer

The

x -

intercept is where the graph of the function crosses the

x -

axis, and the zero of the function is the input value for which

f (x) = 0.

5.3-3 §5.3 source ↗

Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.

Answer

If we evaluate the function at

a

and at

b

and the sign of the function value changes, then we know a zero exists between

a

and

b .

5.3-5 §5.3 source ↗

If the graph of a polynomial just touches the x-axis and then changes direction, what can we conclude about the factored form of the polynomial?

Answer

There will be a factor raised to an even power.

5.3-7 §5.3 source ↗

For the following exercises, find the $x -$ or t-intercepts of the polynomial functions.

$C (t) =$ $3 (t + 2) (t - 3) (t + 5)$

Answer

(- 2, 0),

(3, 0),

(- 5, 0)

5.3-9 §5.3 source ↗

For the following exercises, find the $x -$ or t-intercepts of the polynomial functions.

$C (t) =$ $2 t (t - 3) {(t + 1)}^{2}$

Answer

(3, 0),

(- 1, 0),

(0, 0)

5.3-11 §5.3 source ↗

For the following exercises, find the $x -$ or t-intercepts of the polynomial functions.

$C (t) =$ $4 t^{4} +$ $12 t^{3} -$ $40 t^{2}$

Answer

(0, 0),

(- 5, 0),

(2, 0)

5.3-13 §5.3 source ↗

For the following exercises, find the $x -$ or t-intercepts of the polynomial functions.

$f (x) =$ $x^{3} +$ $x^{2} -$ $20 x$

Answer

(0, 0),

(- 5, 0),

(4, 0)

5.3-15 §5.3 source ↗

For the following exercises, find the $x -$ or t-intercepts of the polynomial functions.

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

Answer

(2, 0),

(- 2, 0),

(- 1, 0)

5.3-17 §5.3 source ↗

For the following exercises, find the $x -$ or t-intercepts of the polynomial functions.

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

Answer

(- 2, 0),

(2, 0),

(\frac{1}{2}, 0)

5.3-19 §5.3 source ↗

For the following exercises, find the $x -$ or t-intercepts of the polynomial functions.

$f (x) =$ $2 x^{4} +$ $6 x^{2} -$ $8$

Answer

(1, 0),

(- 1, 0)

5.3-21 §5.3 source ↗

For the following exercises, find the $x -$ or t-intercepts of the polynomial functions.

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

Answer

(0, 0),

(\sqrt{3}, 0),

(- \sqrt{3}, 0)

5.3-23 §5.3 source ↗

For the following exercises, find the $x -$ or t-intercepts of the polynomial functions.

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

Answer

(0, 0),

(1, 0), (- 1, 0),

(2, 0),

(- 2, 0)

5.3-25 §5.3 source ↗

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

$f (x) = x^{3} - 9 x,$ between $x = 2$ and $x = 4.$

Answer

f (2) = - 10

and

f (4) = 28.

Sign change confirms.

5.3-27 §5.3 source ↗

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

$f (x) = - x^{4} + 4,$ between $x = 1$ and $x = 3$ .

Answer

f (1) = 3

and

f (3) = - 77.

Sign change confirms.

5.3-29 §5.3 source ↗

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

$f (x) =$ $x^{3} -$ $100 x +$ $2,$ between $x = 0.01$ and $x = 0.1$

Answer

f (0.01) =

1.000001

and

f (0.1) =

- 7.999.

Sign change confirms.

5.3-31 §5.3 source ↗

For the following exercises, find the zeros and give the multiplicity of each.

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

Answer

0 with multiplicity 2,

- \frac{3}{2}

with multiplicity 5, 4 with multiplicity 2

5.3-33 §5.3 source ↗

For the following exercises, find the zeros and give the multiplicity of each.

$f (x) =$ $x^{2} (x^{2} + 4 x + 4)$

Answer

0 with multiplicity 2, –2 with multiplicity 2

5.3-35 §5.3 source ↗

For the following exercises, find the zeros and give the multiplicity of each.

$f (x) =$ ${(3 x + 2)}^{5} (x^{2} - 10 x + 25)$

Answer

- \frac{2}{3}

with multiplicity 5, 5 with multiplicity 2

5.3-37 §5.3 source ↗

For the following exercises, find the zeros and give the multiplicity of each.

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

Answer

0 with multiplicity 4, 2 with multiplicity 1, −1 with multiplicity 1

5.3-39 §5.3 source ↗

For the following exercises, find the zeros and give the multiplicity of each.

$f (x) =$ $4 x^{5} -$ $12 x^{4} +$ $9 x^{3}$

Answer

\frac{3}{2}

with multiplicity 2, 0 with multiplicity 3

5.3-41 §5.3 source ↗

For the following exercises, find the zeros and give the multiplicity of each.

$f (x) =$ $4 x^{4} (9 x^{4} - 12 x^{3} + 4 x^{2})$

Answer

0 with multiplicity 6, \frac{2}{3} with multiplicity 2

5.3-43 §5.3 source ↗

For the following exercises, graph the polynomial functions. Note $x -$ and $y -$ intercepts, multiplicity, and end behavior.

$g (x) =$ $(x + 4) {(x - 1)}^{2}$

Answer

x-intercepts, $(1, 0)$ with multiplicity 2, $(- 4, 0)$ with multiplicity 1, $y -$ intercept $(0, 4).$ As $x \to$ $-$ $\infty,$ $f (x) \to$ $-$ $\infty,$ $as x \to$ $\infty,$ $f (x) \to$ $\infty .$

5.3-45 §5.3 source ↗

For the following exercises, graph the polynomial functions. Note $x -$ and $y -$ intercepts, multiplicity, and end behavior.

$k (x) =$ ${(x - 3)}^{3} {(x - 2)}^{2}$

Answer

x-intercepts $(3, 0)$ with multiplicity 3, $(2, 0)$ with multiplicity 2, $y -$ intercept $(0, - 108) .$ As $x \to$ $-$ $\infty,$ $f (x) \to$ $-$ $\infty,$ $as x \to$ $\infty,$ $f (x) \to$ $\infty .$

5.3-47 §5.3 source ↗

For the following exercises, graph the polynomial functions. Note $x -$ and $y -$ intercepts, multiplicity, and end behavior.

$n (x) =$ $-$ $3 x (x + 2) (x - 4)$

Answer

x-intercepts $(0, 0),$ $(- 2, 0),$ $(4, 0)$ with multiplicity 1, $y -$ intercept $(0, 0) .$ As $x \to$ $-$ $\infty,$ $f (x) \to$ $\infty,$ $as x \to$ $\infty,$ $f (x) \to$ $-$ $\infty .$

5.3-49 §5.3 source ↗

For the following exercises, use the graphs to write the formula for a polynomial function of least degree.

Graph of a negative odd-degree polynomial with zeros at x=-3, 1, and 3.

Answer

f (x) =

-

\frac{2}{9} (x - 3) (x + 1) (x + 3)

5.3-51 §5.3 source ↗

For the following exercises, use the graphs to write the formula for a polynomial function of least degree.

Graph of a positive odd-degree polynomial with zeros at x=-2, and 3.

Answer

f (x) =

\frac{1}{4} {(x + 2)}^{2} (x - 3)

5.3-53 §5.3 source ↗

For the following exercises, use the graph to identify zeros and multiplicity.

Graph of a negative even-degree polynomial with zeros at x=-4, -2, 1, and 3.

Answer

–4, –2, 1, 3 with multiplicity 1

5.3-55 §5.3 source ↗

For the following exercises, use the graph to identify zeros and multiplicity.

Graph of a positive even-degree polynomial with zeros at x=-2,, and 3.

Answer

–2, 3 each with multiplicity 2

5.3-57 §5.3 source ↗

For the following exercises, use the given information about the polynomial graph to write the equation.

Degree 3. Zeros at $x = –2,$ $x = 1,$ and $x = 3.$ y-intercept at $(0, –4) .$

Answer

f (x) =

-

\frac{2}{3} (x + 2) (x - 1) (x - 3)

5.3-59 §5.3 source ↗

For the following exercises, use the given information about the polynomial graph to write the equation.

Degree 5. Roots of multiplicity 2 at $x = 3$ and $x = 1$ , and a root of multiplicity 1 at $x = –3.$ y-intercept at $(0, 9)$

Answer

f (x) =

\frac{1}{3} {(x - 3)}^{2} {(x - 1)}^{2} (x + 3)

5.3-61 §5.3 source ↗

For the following exercises, use the given information about the polynomial graph to write the equation.

Degree 5. Double zero at $x = 1,$ and triple zero at $x = 3.$ Passes through the point $(2, 15) .$

Answer

f (x) =

−15 {(x - 1)}^{2} {(x - 3)}^{3}

5.3-63 §5.3 source ↗

For the following exercises, use the given information about the polynomial graph to write the equation.

Degree 3. Zeros at $x = −3,$ $x = −2$ and $x = 1.$ y-intercept at $(0, 12) .$

Answer

f (x) =

-

2 (x + 3) (x + 2) (x - 1)

5.3-65 §5.3 source ↗

For the following exercises, use the given information about the polynomial graph to write the equation.

Degree 4. Roots of multiplicity 2 at $x = \frac{1}{2}$ and roots of multiplicity 1 at $x = 6$ and $x = −2.$

y-intercept at $(0, 18) .$

Answer

f (x) =

-

\frac{3}{2} {(2 x - 1)}^{2} (x - 6) (x + 2)

5.3-67 §5.3 source ↗

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.

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

Answer

local max

(- .58, – .62),

local min

(.58, –1 .38)

5.3-69 §5.3 source ↗

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.

$f (x) = x^{4} + x$

Answer

global min

(- .63, – .47)

5.3-71 §5.3 source ↗

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.

$f (x) =$ $x^{4} -$ $x^{3} +$ $1$

Answer

global min

(.75, .89)

5.3-73 §5.3 source ↗

For the following exercises, use the graphs to write a polynomial function of least degree.

Graph of a positive odd-degree polynomial with zeros at x=--200, and 500 and y=50000000.

Answer

f (x) =

{(x - 500)}^{2} (x + 200)

5.3-75 §5.3 source ↗

For the following exercises, write the polynomial function that models the given situation.

A rectangle has a length of 10 units and a width of 8 units. Squares of $x$ by $x$ units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of $x .$

Answer

f (x) =

4 x^{3} -

36 x^{2} +

80 x

5.3-77 §5.3 source ↗

For the following exercises, write the polynomial function that models the given situation.

A square has sides of 12 units. Squares $x + 1$ by $x + 1$ units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a function in terms of $x .$

Answer

f (x) =

4 x^{3} -

36 x^{2} +

60 x +

100

5.3-79 §5.3 source ↗

For the following exercises, write the polynomial function that models the given situation.

A right circular cone has a radius of $3 x + 6$ and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is $V = \frac{1}{3} π r^{2} h$ for radius $r$ and height $h .$

Answer

f (x) =

9 π (x^{3} + 5 x^{2} + 8 x + 4)

Try It #1 §5.4 source ↗

Divide

16 x^{3} -

12 x^{2} +

20 x -

3

4 x + 5.

Answer

4 x^{2} -

8 x +

15 -

\frac{78}{4 x + 5}

Try It #2 §5.4 source ↗

Use synthetic division to divide

3 x^{4} +

18 x^{3} -

3 x +

40

x + 7.

Answer

3 x^{3} -

3 x^{2} +

21 x -

150 +

\frac{1, 090}{x + 7}

Try It #3 §5.4 source ↗

The area of a rectangle is given by

3 x^{3} +

14 x^{2} -

23 x +

6.

The width of the rectangle is given by

x + 6.

Find an expression for the length of the rectangle.

Answer

3 x^{2} - 4 x + 1

5.4-1 §5.4 source ↗

If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?

Answer

The binomial is a factor of the polynomial.

5.4-3 §5.4 source ↗

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(x^{2} + 5 x - 1) \div (x - 1)$

Answer

x +

6 +

\frac{5}{x - 1}, quotient: x +

6, remainder: 5

5.4-5 §5.4 source ↗

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(3 x^{2} + 23 x + 14) \div (x + 7)$

Answer

3 x +

2, quotient: 3 x +

2, remainder: 0

5.4-7 §5.4 source ↗

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(6 x^{2} - 25 x - 25) \div (6 x + 5)$

Answer

x -

5, quotient: x -

5, remainder: 0

5.4-9 §5.4 source ↗

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(2 x^{2} - 3 x + 2) \div (x + 2)$

Answer

2 x -

7 +

\frac{16}{x + 2}, quotient: ​ 2 x -

7, remainder: 16

5.4-11 §5.4 source ↗

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(3 x^{2} - 5 x + 4) \div (3 x + 1)$

Answer

x -

2 +

\frac{6}{3 x + 1}, quotient: x -

2, remainder: 6

5.4-13 §5.4 source ↗

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(2 x^{3} + 3 x^{2} - 4 x + 15) \div (x + 3)$

Answer

2 x^{2} -

3 x +

5, quotient: 2 x^{2} -

3 x +

5, remainder: 0

5.4-15 §5.4 source ↗

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)

$(2 x^{3} - 6 x^{2} - 7 x + 6) \div (x - 4)$

Answer

2 x^{2} +

2 x +

1 +

\frac{10}{x - 4}

5.4-17 §5.4 source ↗

$(4 x^{3} - 12 x^{2} - 5 x - 1) \div (2 x + 1)$

Answer

2 x^{2} -

7 x +

1 -

\frac{2}{2 x + 1}

5.4-19 §5.4 source ↗

$(3 x^{3} - 2 x^{2} + x - 4) \div (x + 3)$

Answer

3 x^{2} -

11 x +

34 -

\frac{106}{x + 3}

5.4-21 §5.4 source ↗

$(2 x^{3} + 7 x^{2} - 13 x - 3) \div (2 x - 3)$

Answer

x^{2} + 5 x + 1

5.4-23 §5.4 source ↗

$(4 x^{3} - 5 x^{2} + 13) \div (x + 4)$

Answer

4 x^{2} -

21 x +

84 -

\frac{323}{x + 4}

5.4-25 §5.4 source ↗

$(x^{3} - 21 x^{2} + 147 x - 343) \div (x - 7)$

Answer

x^{2} - 14 x + 49

5.4-27 §5.4 source ↗

$(9 x^{3} - x + 2) \div (3 x - 1)$

Answer

3 x^{2} + x + \frac{2}{3 x - 1}

5.4-29 §5.4 source ↗

$(x^{4} + x^{3} - 3 x^{2} - 2 x + 1) \div (x + 1)$

Answer

x^{3} - 3 x + 1

5.4-31 §5.4 source ↗

$(x^{4} + 2 x^{3} - 3 x^{2} + 2 x + 6) \div (x + 3)$

Answer

x^{3} - x^{2} + 2

5.4-33 §5.4 source ↗

$(x^{4} - 8 x^{3} + 24 x^{2} - 32 x + 16) \div (x - 2)$

Answer

x^{3} -

6 x^{2} +

12 x -

8

5.4-35 §5.4 source ↗

$(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) \div (x - 3)$

Answer

x^{3} -

9 x^{2} +

27 x -

27

5.4-37 §5.4 source ↗

$(4 x^{4} + 2 x^{3} - 4 x^{2} + 2 x + 2) \div (2 x + 1)$

Answer

2 x^{3} - 2 x + 2

5.4-39 §5.4 source ↗

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

$x -$ $2,$ $3 x^{4} -$ $6 x^{3} -$ $5 x +$ $10$

Answer

Yes

(x - 2) (3 x^{3} - 5)

5.4-41 §5.4 source ↗

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

$x -$ $2,$ $4 x^{4} -$ $15 x^{2} -$ $4$

Answer

Yes

(x - 2) (4 x^{3} + 8 x^{2} + x + 2)

5.4-43 §5.4 source ↗

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

$x +$ $\frac{1}{3},$ $3 x^{4} +$ $x^{3} -$ $3 x +$ $1$

Answer

5.4-45 §5.4 source ↗

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.

Factor is $(x^{2} + 2 x + 4)$

Graph of a polynomial that has a x-intercept at 1.

Answer

(x - 1) (x^{2} + 2 x + 4)

5.4-47 §5.4 source ↗

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.

Factor is $x^{2} + x + 1$

Graph of a polynomial that has a x-intercept at 5.

Answer

(x - 5) (x^{2} + x + 1)

5.4-49 §5.4 source ↗

For the following exercises, use synthetic division to find the quotient and remainder.

$\frac{4 x^{3} - 33}{x - 2}$

Answer

Quotient: 4 x^{2} +

8 x +

16, remainder: -

1

5.4-51 §5.4 source ↗

For the following exercises, use synthetic division to find the quotient and remainder.

$\frac{3 x^{3} + 2 x - 5}{x - 1}$

Answer

Quotient: 3 x^{2} +

3 x +

5, remainder: 0

5.4-53 §5.4 source ↗

For the following exercises, use synthetic division to find the quotient and remainder.

$\frac{x^{4} - 22}{x + 2}$

Answer

Quotient: x^{3} -

2 x^{2} +

4 x -

8, remainder: -

6

5.4-55 §5.4 source ↗

For the following exercises, use a calculator with CAS to answer the questions.

Consider $\frac{x^{k} + 1}{x + 1}$ for $k =$ $1,$ $3,$ $5.$ What do you expect the result to be if $k = 7 ?$

Answer

x^{6} -

x^{5} +

x^{4} -

x^{3} +

x^{2} -

x +

1

5.4-57 §5.4 source ↗

For the following exercises, use a calculator with CAS to answer the questions.

Consider $\frac{x^{k}}{x + 1}$ with $k =$ $1,$ $2,$ $3.$ What do you expect the result to be if $k = 4 ?$

Answer

x^{3} -

x^{2} +

x -

1 +

\frac{1}{x + 1}

5.4-59 §5.4 source ↗

For the following exercises, use synthetic division to determine the quotient involving a complex number.

$\frac{x + 1}{x - i}$

Answer

1 + \frac{1 + i}{x - i}

5.4-61 §5.4 source ↗

For the following exercises, use synthetic division to determine the quotient involving a complex number.

$\frac{x + 1}{x + i}$

Answer

1 + \frac{1 - i}{x + i}

5.4-63 §5.4 source ↗

For the following exercises, use synthetic division to determine the quotient involving a complex number.

$\frac{x^{3} + 1}{x - i}$

Answer

x^{2} +

i x -

1 +

\frac{1 - i}{x - i}

5.4-65 §5.4 source ↗

For the following exercises, use the given length and area of a rectangle to express the width algebraically.

Length is $2 x + 5,$ area is $4 x^{3} +$ $10 x^{2} +$ $6 x +$ $15$

Answer

2 x^{2} + 3

5.4-67 §5.4 source ↗

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.

Volume is $12 x^{3} +$ $20 x^{2} -$ $21 x -$ $36,$ length is $2 x + 3,$ width is $3 x - 4.$

Answer

2 x + 3

5.4-69 §5.4 source ↗

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.

Volume is $10 x^{3} +$ $27 x^{2} +$ $2 x -$ $24,$ length is $5 x - 4,$ width is $2 x + 3.$

Answer

x + 2

5.4-71 §5.4 source ↗

For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.

Volume is $π (25 x^{3} - 65 x^{2} - 29 x - 3),$ radius is $5 x + 1.$

Answer

x - 3

5.4-73 §5.4 source ↗

For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.

Volume is $π (3 x^{4} + 24 x^{3} + 46 x^{2} - 16 x - 32),$ radius is $x + 4.$

Answer

3 x^{2} - 2

Try It #1 §5.5 source ↗

Use the Remainder Theorem to evaluate

f (x) =

2 x^{5} -

3 x^{4} -

9 x^{3} +

8 x^{2} +

2

x = - 3.

Answer

f (- 3) = - 412

Try It #2 §5.5 source ↗

Use the Factor Theorem to find the zeros of

f (x) =

x^{3} +

4 x^{2} -

4 x -

16

given that

(x - 2)

is a factor of the polynomial.

Answer

The zeros are 2, –2, and –4.

Try It #3 §5.5 source ↗

Use the Rational Zero Theorem to find the rational zeros of

f (x) =

x^{3} -

5 x^{2} +

2 x +

1.

Answer

There are no rational zeros.

Try It #4 §5.5 source ↗

Find the zeros of

f (x) =

2 x^{3} +

5 x^{2} -

11 x +

4.

Answer

The zeros are

–4, \frac{1}{2},

and 1 .

Try It #5 §5.5 source ↗

Find a third degree polynomial with real coefficients that has zeros of 5 and

- 2 i

such that

f (1) = 10.

Answer

f (x) =

-

\frac{1}{2} x^{3} +

\frac{5}{2} x^{2} -

2 x +

10

Try It #6 §5.5 source ↗

Use Descartes’ Rule of Signs to determine the maximum possible numbers of positive and negative real zeros for

f (x) =

2 x^{4} -

10 x^{3} +

11 x^{2} -

15 x +

12.

Use a graph to verify the numbers of positive and negative real zeros for the function.

Answer

There must be 4, 2, or 0 positive real roots and 0 negative real roots. The graph shows that there are 2 positive real zeros and 0 negative real zeros.

Try It #7 §5.5 source ↗

A shipping container in the shape of a rectangular solid must have a volume of 84 cubic meters. The client tells the manufacturer that, because of the contents, the length of the container must be one meter longer than the width, and the height must be one meter greater than twice the width. What should the dimensions of the container be?

Answer

3 meters by 4 meters by 7 meters

5.5-1 §5.5 source ↗

Describe a use for the Remainder Theorem.

Answer

The theorem can be used to evaluate a polynomial.

5.5-3 §5.5 source ↗

What is the difference between rational and real zeros?

Answer

Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.

5.5-5 §5.5 source ↗

If synthetic division reveals a zero, why should we try that value again as a possible solution?

Answer

Polynomial functions can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.

5.5-7 §5.5 source ↗

For the following exercises, use the Remainder Theorem to find the remainder.

$(3 x^{3} - 2 x^{2} + x - 4) \div (x + 3)$

Answer

- 106

5.5-9 §5.5 source ↗

For the following exercises, use the Remainder Theorem to find the remainder.

$(- 3 x^{2} + 6 x + 24) \div (x - 4)$

Answer

0

5.5-11 §5.5 source ↗

For the following exercises, use the Remainder Theorem to find the remainder.

$(x^{4} - 1) \div (x - 4)$

Answer

255

5.5-13 §5.5 source ↗

For the following exercises, use the Remainder Theorem to find the remainder.

$(4 x^{3} + 5 x^{2} - 2 x + 7) \div (x + 2)$

Answer

- 1

5.5-15 §5.5 source ↗

For the following exercises, use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.

$f (x) =$ $2 x^{3} +$ $x^{2} -$ $5 x +$ $2;$ $x +$ $2$

Answer

-

2,

1,

\frac{1}{2}

5.5-17 §5.5 source ↗

For the following exercises, use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.

$f (x) =$ $2 x^{3} +$ $3 x^{2} +$ $x +$ $6;$ $x +$ $2$

Answer

- 2

5.5-19 §5.5 source ↗

For the following exercises, use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.

$x^{3} +$ $3 x^{2} +$ $4 x +$ $12;$ $x +$ $3$

Answer

- 3

5.5-21 §5.5 source ↗

For the following exercises, use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.

$2 x^{3} +$ $5 x^{2} -$ $12 x -$ $30,$ $ 2 x +$ $5$

Answer

-

\frac{5}{2},

\sqrt{6},

-

\sqrt{6}

5.5-23 §5.5 source ↗

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.

$2 x^{3} +$ $7 x^{2} -$ $10 x -$ $24 =$ $0$

Answer

2,

-

4,

-

\frac{3}{2}

5.5-25 §5.5 source ↗

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.

$x^{3} +$ $5 x^{2} -$ $16 x -$ $80 =$ $0$

Answer

4,

-

4,

-

5

5.5-27 §5.5 source ↗

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.

$2 x^{3} -$ $3 x^{2} -$ $32 x -$ $15 =$ $0$

Answer

5,

-

3,

-

\frac{1}{2}

5.5-29 §5.5 source ↗

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.

$2 x^{3} -$ $3 x^{2} -$ $x +$ $1 =$ $0$

Answer

\frac{1}{2},

\frac{1 + \sqrt{5}}{2},

\frac{1 - \sqrt{5}}{2}

5.5-31 §5.5 source ↗

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.

$2 x^{3} -$ $5 x^{2} +$ $9 x -$ $9 =$ $0$

Answer

\frac{3}{2}

5.5-33 §5.5 source ↗

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.

$x^{4} -$ $2 x^{3} -$ $7 x^{2} +$ $8 x +$ $12 =$ $0$

Answer

2,

3,

-

1,

-

2

5.5-35 §5.5 source ↗

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.

$4 x^{4} +$ $4 x^{3} -$ $25 x^{2} -$ $x +$ $6 =$ $0$

Answer

\frac{1}{2},

-

\frac{1}{2},

2,

-

3

5.5-37 §5.5 source ↗

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.

$x^{4} +$ $2 x^{3} -$ $4 x^{2} -$ $10 x -$ $5 =$ $0$

Answer

-

1,

-

1,

\sqrt{5},

-

\sqrt{5}

5.5-39 §5.5 source ↗

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.

$8 x^{^{4}} +$ $26 x^{3} +$ $39 x^{2} +$ $26 x +$ $6$

Answer

-

\frac{3}{4},

-

\frac{1}{2}

5.5-41 §5.5 source ↗

For the following exercises, find all complex solutions (real and non-real).

$x^{3} -$ $8 x^{2} +$ $25 x -$ $26 =$ $0$

Answer

2,

3 +

2 i,

3 -

2 i

5.5-43 §5.5 source ↗

For the following exercises, find all complex solutions (real and non-real).

$3 x^{3} -$ $4 x^{2} +$ $11 x +$ $10 =$ $0$

Answer

-

\frac{2}{3},

1 +

2 i,

1 -

2 i

5.5-45 §5.5 source ↗

For the following exercises, find all complex solutions (real and non-real).

$2 x^{3} -$ $3 x^{2} +$ $32 x +$ $17 =$ $0$

Answer

-

\frac{1}{2},

1 +

4 i,

1 -

4 i

5.5-47 §5.5 source ↗

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.

$f (x) =$ $x^{4} -$ $x^{2} -$ $1$

Answer

1 positive, 1 negative

5.5-49 §5.5 source ↗

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.

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

Answer

3 or 1 positive, 0 negative

5.5-51 §5.5 source ↗

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.

$f (x) =$ $2 x^{3} +$ $37 x^{2} +$ $200 x +$ $300$

Answer

0 positive, 3 or 1 negative

5.5-53 §5.5 source ↗

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.

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

Answer

2 or 0 positive, 2 or 0 negative

5.5-55 §5.5 source ↗

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.

$f (x) =$ $10 x^{4} -$ $21 x^{2} +$ $11$

Answer

2 or 0 positive, 2 or 0 negative

5.5-57 §5.5 source ↗

For the following exercises, list all possible rational zeros for the functions.

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

Answer

\pm 5,

\pm 1,

\pm \frac{5}{2},

\pm \frac{1}{2}

5.5-59 §5.5 source ↗

For the following exercises, list all possible rational zeros for the functions.

$f (x) =$ $6 x^{4} -$ $10 x^{2} +$ $13 x +$ $1$

Answer

\pm 1,

\pm \frac{1}{2},

\pm \frac{1}{3},

\pm \frac{1}{6}

5.5-61 §5.5 source ↗

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

$f (x) =$ $6 x^{3} -$ $7 x^{2} +$ $1$

Answer

1,

\frac{1}{2},

-

\frac{1}{3}

5.5-63 §5.5 source ↗

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

$f (x) =$ $8 x^{3} -$ $6 x^{2} -$ $23 x +$ $6$

Answer

2,

\frac{1}{4},

-

\frac{3}{2}

5.5-65 §5.5 source ↗

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

$f (x) =$ $16 x^{4} -$ $24 x^{3} +$ $x^{2} -$ $15 x +$ $25$

Answer

\frac{5}{4}

5.5-67 §5.5 source ↗

For the following exercises, construct a polynomial function of least degree possible using the given information.

Real roots: –1, 1 (with multiplicity 2 and 1) and $(2, f (2)) =$ $(2, 4)$

Answer

f (x) =

\frac{4}{9} (x^{3} + x^{2} - x - 1)

5.5-69 §5.5 source ↗

For the following exercises, construct a polynomial function of least degree possible using the given information.

Real roots: $- \frac{1}{2}$ , 0, $\frac{1}{2}$ and $(- 2, f (- 2)) =$ $(- 2, 6)$

Answer

f (x) =

-

\frac{1}{5} (4 x^{3} - x)

5.5-71 §5.5 source ↗

For the following exercises, find the dimensions of the box described.

The length is twice as long as the width. The height is 2 inches greater than the width. The volume is 192 cubic inches.

Answer

8 by 4 by 6 inches

5.5-73 §5.5 source ↗

For the following exercises, find the dimensions of the box described.

The length is one inch more than the width, which is one inch more than the height. The volume is 86.625 cubic inches.

Answer

5.5 by 4.5 by 3.5 inches

5.5-75 §5.5 source ↗

For the following exercises, find the dimensions of the box described.

The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.

Answer

8 by 5 by 3 inches

5.5-77 §5.5 source ↗

For the following exercises, find the dimensions of the right circular cylinder described.

The height is one less than one half the radius. The volume is $72 π$ cubic meters.

Answer

Radius = 6 meters, Height = 2 meters

5.5-79 §5.5 source ↗

For the following exercises, find the dimensions of the right circular cylinder described.

The radius and height differ by two meters. The height is greater and the volume is $28.125 π$ cubic meters.

Answer

Radius = 2.5 meters, Height = 4.5 meters

Try It #1 §5.6 source ↗

Use arrow notation to describe the end behavior and local behavior for the reciprocal squared function.

Answer

End behavior: as

x \to

\pm \infty,

f (x) \to

0;

Local behavior: as

x \to

0,

f (x) \to

\infty

(there are no x- or y-intercepts)

Try It #2 §5.6 source ↗

Sketch the graph, and find the horizontal and vertical asymptotes of the reciprocal squared function that has been shifted right 3 units and down 4 units.

Answer

Graph of f(x)=1/(x-3)^2-4 with its vertical asymptote at x=3 and its horizontal asymptote at y=-4.

The function and the asymptotes are shifted 3 units right and 4 units down. As $x \to 3, f (x) \to \infty,$ and as $x \to$ $\pm \infty,$ $f (x) \to$ $-$ $4.$

The function is $f (x) =$ $\frac{1}{{(x - 3)}^{2}} -$ $4.$

Try It #3 §5.6 source ↗

There are 1,200 first-year and 1,500 second-year students at a rally at noon. After 12 p.m., 20 first-year students arrive at the rally every five minutes while 15 second-year students leave the rally. Find the ratio of first-year to second-year students at 1 p.m.

Answer

\frac{12}{11}

Try It #4 §5.6 source ↗

Find the domain of

f (x) =

\frac{4 x}{5 (x - 1) (x - 5)} .

Answer

The domain is all real numbers except

x = 1

and

x = 5.

Try It #5 §5.6 source ↗

Find the vertical asymptotes and removable discontinuities of the graph of

f (x) =

\frac{x^{2} - 25}{x^{3} - 6 x^{2} + 5 x} .

Answer

Removable discontinuity at

x = 5.

Vertical asymptotes:

x = 0, x = 1.

Try It #6 §5.6 source ↗

Find the vertical and horizontal asymptotes of the function:

$f (x) =$ $\frac{(2 x - 1) (2 x + 1)}{(x - 2) (x + 3)}$

Answer

Vertical asymptotes at

x = 2

and

x = - 3;

horizontal asymptote at

y = 4.

Try It #7 §5.6 source ↗

Given the reciprocal squared function that is shifted right 3 units and down 4 units, write this as a rational function. Then, find the x- and y-intercepts and the horizontal and vertical asymptotes.

Answer

For the transformed reciprocal squared function, we find the rational form. $f (x) =$ $\frac{1}{{(x - 3)}^{2}} -$ $4 =$ $\frac{1 - 4 {(x - 3)}^{2}}{{(x - 3)}^{2}} =$ $\frac{1 - 4 (x^{2} - 6 x + 9)}{(x - 3) (x - 3)} =$ $\frac{- 4 x^{2} + 24 x - 35}{x^{2} - 6 x + 9}$

Because the numerator is the same degree as the denominator we know that as $x \to$ $\pm \infty,$ $f (x) \to$ $-$ $4;$ $so y =$ $- 4$ is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is $x = 3,$ because as $x \to 3, f (x) \to \infty .$ We then set the numerator equal to 0 and find the x-intercepts are at $(2.5, 0)$ and $(3.5, 0) .$ Finally, we evaluate the function at 0 and find the y-intercept to be at $(0, \frac{- 35}{9}) .$

Try It #8 §5.6 source ↗

Given the function

f (x) =

\frac{{(x + 2)}^{2} (x - 2)}{2 {(x - 1)}^{2} (x - 3)},

use the characteristics of polynomials and rational functions to describe its behavior and sketch the function.

Answer

Horizontal asymptote at $y = \frac{1}{2} .$ Vertical asymptotes at $x =$ $1 and x =$ $3.$ y-intercept at $(0, \frac{4}{3} .)$

x-intercepts at $(2, 0) and (- 2, 0) .$ $(- 2, 0)$ is a zero with multiplicity 2, and the graph bounces off the x-axis at this point. $(2, 0)$ is a single zero and the graph crosses the axis at this point.

Graph of f(x)=(x+2)^2(x-2)/2(x-1)^2(x-3) with its vertical and horizontal asymptotes.

5.6-1 §5.6 source ↗

What is the fundamental difference in the algebraic representation of a polynomial function and a rational function?

Answer

The rational function will be represented by a quotient of polynomial functions.

5.6-3 §5.6 source ↗

If the graph of a rational function has a removable discontinuity, what must be true of the functional rule?

Answer

The numerator and denominator must have a common factor.

5.6-5 §5.6 source ↗

Can a graph of a rational function have no x-intercepts? If so, how?

Answer

Yes. The numerator of the formula of the functions would have only complex roots and/or factors common to both the numerator and denominator.

5.6-7 §5.6 source ↗

For the following exercises, find the domain of the rational functions.

$f (x) =$ $\frac{x + 1}{x^{2} - 1}$

Answer

All reals x \neq - 1,

1

5.6-9 §5.6 source ↗

For the following exercises, find the domain of the rational functions.

$f (x) =$ $\frac{x^{2} + 4 x - 3}{x^{4} - 5 x^{2} + 4}$

Answer

All reals x \neq - 1,

- 2,

1,

2

5.6-11 §5.6 source ↗

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

$f (x) = \frac{2}{5 x + 2}$

Answer

V.A. at

x = - \frac{2}{5};

H.A. at

y = 0;

Domain is all reals

x \neq - \frac{2}{5}

5.6-13 §5.6 source ↗

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

$f (x) =$ $\frac{x}{x^{2} + 5 x - 36}$

Answer

V.A. at

x =

4,

- 9;

H.A. at

y = 0;

Domain is all reals

x \neq 4,

- 9

5.6-15 §5.6 source ↗

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

$f (x) =$ $\frac{3 x - 4}{x^{3} - 16 x}$

Answer

V.A. at

x =

0,

4,

-

4;

H.A. at

y = 0;

Domain is all reals

x \neq 0,

4,

- 4

5.6-17 §5.6 source ↗

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

$f (x) =$ $\frac{x + 5}{x^{2} - 25}$

Answer

V.A. at

x = 5;

H.A. at

y = 0;

Domain is all reals

x \neq 5, - 5

5.6-19 §5.6 source ↗

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

$f (x) =$ $\frac{4 - 2 x}{3 x - 1}$

Answer

V.A. at

x = \frac{1}{3};

H.A. at

y = - \frac{2}{3};

Domain is all reals

x \neq \frac{1}{3} .

5.6-21 §5.6 source ↗

For the following exercises, find the x- and y-intercepts for the functions.

$f (x) = \frac{x}{x^{2} - x}$

Answer

none

5.6-23 §5.6 source ↗

For the following exercises, find the x- and y-intercepts for the functions.

$f (x) =$ $\frac{x^{2} + x + 6}{x^{2} - 10 x + 24}$

Answer

x -intercepts none, y -intercept (0, \frac{1}{4})

5.6-25 §5.6 source ↗

For the following exercises, describe the local and end behavior of the functions.

$f (x) = \frac{x}{2 x + 1}$

Answer

Local behavior: $x \to$ $-$ ${\frac{1}{2}}^{+},$ $f (x) \to$ $-$ $\infty,$ $x \to$ $-$ ${\frac{1}{2}}^{-},$ $f (x) \to$ $\infty$

End behavior: $x \to$ $\pm \infty,$ $f (x) \to$ $\frac{1}{2}$

5.6-27 §5.6 source ↗

For the following exercises, describe the local and end behavior of the functions.

$f (x) = \frac{- 2 x}{x - 6}$

Answer

Local behavior:

x \to

6^{+},

f (x) \to

-

\infty,

x \to

6^{-},

f (x) \to

\infty,

End behavior:

x \to

\pm \infty,

f (x) \to

-

2

5.6-29 §5.6 source ↗

For the following exercises, describe the local and end behavior of the functions.

$f (x) =$ $\frac{2 x^{2} - 32}{6 x^{2} + 13 x - 5}$

Answer

Local behavior: $x \to$ ${\frac{1}{3}}^{+},$ $f (x) \to$ $-$ $\infty,$ $x \to$ ${\frac{1}{3}}^{-},$ $f (x) \to$ $\infty,$ $x \to$ ${- \frac{5}{2}}^{+},$ $f (x) \to$ $\infty,$ $x \to$ ${- \frac{5}{2}}^{-},$ $f (x) \to$ $-$ $\infty$

End behavior: $x \to$ $\pm \infty,$ $f (x) \to$ $\frac{1}{3}$

5.6-31 §5.6 source ↗

For the following exercises, find the slant asymptote of the functions.

$f (x) =$ $\frac{4 x^{2} - 10}{2 x - 4}$

Answer

y = 2 x + 4

5.6-33 §5.6 source ↗

For the following exercises, find the slant asymptote of the functions.

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

Answer

y = 2 x

5.6-35 §5.6 source ↗

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes.

The reciprocal function shifted up two units.

Answer

$V . A . x =$ $0,$ $H . A . y =$ $2$

5.6-37 §5.6 source ↗

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes.

The reciprocal squared function shifted to the right 2 units.

Answer

$V . A . x =$ $2,$ $H . A . y =$ $0$

5.6-39 §5.6 source ↗

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph.

$p (x) =$ $\frac{2 x - 3}{x + 4}$

Answer

$V . A . x =$ $-$ $4,$ $H . A . y =$ $2;$ $(\frac{3}{2}, 0);$ $(0, - \frac{3}{4})$

Graph of p(x)=(2x-3)/(x+4) with its vertical asymptote at x=-4 and horizontal asymptote at y=2.

5.6-41 §5.6 source ↗

$s (x) =$ $\frac{4}{{(x - 2)}^{2}}$

Answer

$V . A . x =$ $2,$ $H . A . y =$ $0,$ $(0, 1)$

Graph of s(x)=4/(x-2)^2 with its vertical asymptote at x=2 and horizontal asymptote at y=0.

5.6-43 §5.6 source ↗

$f (x) =$ $\frac{3 x^{2} - 14 x - 5}{3 x^{2} + 8 x - 16}$

Answer

$V . A . x =$ $-$ $4,$ $x =$ $\frac{4}{3},$ $H . A . y =$ $1;$ $(5, 0);$ $(- \frac{1}{3}, 0);$ $(0, \frac{5}{16})$

Graph of f(x)=(3x^2-14x-5)/(3x^2+8x-16) with its vertical asymptotes at x=-4 and x=4/3 and horizontal asymptote at y=1.

5.6-45 §5.6 source ↗

$a (x) =$ $\frac{x^{2} + 2 x - 3}{x^{2} - 1}$

Answer

$V . A . x =$ $-$ $1,$ $H . A . y =$ $1;$ $(- 3, 0);$ $(0, 3)$

Graph of a(x)=(x^2+2x-3)/(x^2-1) with its vertical asymptote at x=-1 and horizontal asymptote at y=1.

5.6-47 §5.6 source ↗

$h (x) =$ $\frac{2 x^{2} + x - 1}{x - 4}$

Answer

$V . A . x =$ $4,$ $S . A . y =$ $2 x +$ $9;$ $(- 1, 0);$ $(\frac{1}{2}, 0);$ $(0, \frac{1}{4})$

Graph of h(x)=(2x^2+x-1)/(x-1) with its vertical asymptote at x=4 and slant asymptote at y=2x+9.

5.6-49 §5.6 source ↗

$w (x) =$ $\frac{(x - 1) (x + 3) (x - 5)}{{(x + 2)}^{2} (x - 4)}$

Answer

$V . A . x =$ $-$ $2,$ $x =$ $4,$ $H . A . y =$ $1,$ $(1, 0);$ $(5, 0);$ $(- 3, 0);$ $(0, - \frac{15}{16})$

Graph of w(x)=(x-1)(x+3)(x-5)/(x+2)^2(x-4) with its vertical asymptotes at x=-2 and x=4 and horizontal asymptote at y=1.

5.6-51 §5.6 source ↗

For the following exercises, write an equation for a rational function with the given characteristics.

Vertical asymptotes at $x = 5$ and $x = −5,$ x-intercepts at $(2, 0)$ and $(−1, 0),$ y-intercept at $(0, 4)$

Answer

y =

50 \frac{x^{2} - x - 2}{x^{2} - 25}

5.6-53 §5.6 source ↗

For the following exercises, write an equation for a rational function with the given characteristics.

Vertical asymptotes at $x = −4$ and $x = −5,$ x-intercepts at $(4, 0)$ and $(−6, 0),$ Horizontal asymptote at $y = 7$

Answer

y =

7 \frac{x^{2} + 2 x - 24}{x^{2} + 9 x + 20}

5.6-55 §5.6 source ↗

For the following exercises, write an equation for a rational function with the given characteristics.

Vertical asymptote at $x = −1,$ Double zero at $x = 2,$ y-intercept at $(0, 2)$

Answer

y =

\frac{1}{2} \frac{x^{2} - 4 x + 4}{x + 1}

5.6-57 §5.6 source ↗

For the following exercises, use the graphs to write an equation for the function.

Graph of a rational function with vertical asymptotes at x=-3 and x=4.

Answer

y =

4 \frac{x - 3}{x^{2} - x - 12}

5.6-59 §5.6 source ↗

For the following exercises, use the graphs to write an equation for the function.

Graph of a rational function with vertical asymptotes at x=-3 and x=3.

Answer

y =

\frac{27 (x - 2)}{(x + 3) {(x - 3)}^{2}}

5.6-61 §5.6 source ↗

For the following exercises, use the graphs to write an equation for the function.

Graph of a rational function with vertical asymptote at x=1.

Answer

y =

\frac{1}{3} \frac{x^{2} + x - 6}{x - 1}

5.6-63 §5.6 source ↗

For the following exercises, use the graphs to write an equation for the function.

Graph of a rational function with vertical asymptotes at x=-3 and x=2.

Answer

y =

-

6 \frac{{(x - 1)}^{2}}{(x + 3) {(x - 2)}^{2}}

5.6-65 §5.6 source ↗

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote

$f (x) = \frac{1}{x - 2}$

Answer

$x$	2.01	2.001	2.0001	1.99	1.999
$y$	100	1,000	10,000	–100	–1,000

$x$	10	100	1,000	10,000	100,000
$y$	.125	.0102	.001	.0001	.00001

Vertical asymptote $x = 2,$ Horizontal asymptote $y = 0$

5.6-67 §5.6 source ↗

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote

$f (x) = \frac{2 x}{x + 4}$

Answer

$x$	–4.1	–4.01	–4.001	–3.99	–3.999
$y$	82	802	8,002	–798	–7998

$x$	10	100	1,000	10,000	100,000
$y$	1.4286	1.9331	1.992	1.9992	1.999992

Vertical asymptote $x = - 4,$ Horizontal asymptote $y = 2$

5.6-69 §5.6 source ↗

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote

$f (x) =$ $\frac{x^{2}}{x^{2} + 2 x + 1}$

Answer

$x$	–.9	–.99	–.999	–1.1	–1.01
$y$	81	9,801	998,001	121	10,201

$x$	10	100	1,000	10,000	100,000
$y$	.82645	.9803	.998	.9998

Vertical asymptote $x = - 1,$ Horizontal asymptote $y = 1$

5.6-71 §5.6 source ↗

For the following exercises, use a calculator to graph $f (x) .$ Use the graph to solve $f (x) > 0.$

$f (x) = \frac{4}{2 x - 3}$

Answer

$(\frac{3}{2}, \infty)$

5.6-73 §5.6 source ↗

For the following exercises, use a calculator to graph $f (x) .$ Use the graph to solve $f (x) > 0.$

$f (x) =$ $\frac{x + 2}{(x - 1) (x - 4)}$

Answer

$(- 2, 1) \cup (4, \infty)$

5.6-75 §5.6 source ↗

For the following exercises, identify the removable discontinuity.

$f (x) =$ $\frac{x^{2} - 4}{x - 2}$

Answer

(2, 4)

5.6-77 §5.6 source ↗

For the following exercises, identify the removable discontinuity.

$f (x) =$ $\frac{x^{2} + x - 6}{x - 2}$

Answer

(2, 5)

5.6-79 §5.6 source ↗

For the following exercises, identify the removable discontinuity.

$f (x) =$ $\frac{x^{3} + x^{2}}{x + 1}$

Answer

(- 1, 1)

5.6-81 §5.6 source ↗

For the following exercises, express a rational function that describes the situation.

In the refugee camp hospital, a large mixing tank currently contains 300 gallons of water, into which 8 pounds of sugar have been mixed. A tap will open, pouring 20 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after $t$ minutes.

Answer

C (t) =

\frac{8 + 2 t}{300 + 20 t}

5.6-83 §5.6 source ↗

For the following exercises, use the given rational function to answer the question.

The concentration $C$ of a drug in a patient’s bloodstream_$t$hours after injection is given by $C (t) =$ $\frac{100 t}{2 t^{2} + 75} .$ Use a calculator to approximate the time when the concentration is highest.

Answer

After about 6.12 hours.

5.6-85 §5.6 source ↗

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.

A rectangular box with a square base is to have a volume of 20 cubic feet. The material for the base costs 30 cents/ square foot. The material for the sides costs 10 cents/square foot. The material for the top costs 20 cents/square foot. Determine the dimensions that will yield minimum cost. Let $x$ = length of the side of the base.

Answer

A (x) =

50 x^{2} +

\frac{800}{x} .

2 by 2 by 5 feet.

5.6-87 §5.6 source ↗

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.

A right circular cylinder with no top has a volume of 50 cubic meters. Find the radius that will yield minimum surface area. Let $x$ = radius.

Answer

A (x) =

π x^{2} +

\frac{100}{x} .

Radius = 2.52 meters.

Try It #1 §5.7 source ↗

Show that

f (x) = \frac{x + 5}{3}

and

f^{- 1} (x) = 3 x - 5

are inverses.

Answer

f^{- 1} (f (x)) =

f^{- 1} (\frac{x + 5}{3}) =

3 (\frac{x + 5}{3}) -

5 =

(x - 5) +

5 =

x

and

f (f^{- 1} (x)) =

f (3 x - 5) =

\frac{(3 x - 5) + 5}{3} =

\frac{3 x}{3} =

x

Try It #2 §5.7 source ↗

Find the inverse function of

f (x) = \sqrt[3]{x + 4} .

Answer

f^{- 1} (x) = x^{3} - 4

Try It #3 §5.7 source ↗

Find the inverse of the function

f (x) = x^{2} + 1,

on the domain

x \geq 0.

Answer

f^{- 1} (x) = \sqrt{x - 1}

Try It #4 §5.7 source ↗

Restrict the domain and then find the inverse of the function

f (x) = \sqrt{2 x + 3} .

Answer

f^{- 1} (x) =

\frac{x^{2} - 3}{2},

x \geq 0

Try It #5 §5.7 source ↗

Find the inverse of the function

f (x) =

\frac{x + 3}{x - 2} .

Answer

f^{- 1} (x) =

\frac{2 x + 3}{x - 1}

5.7-1 §5.7 source ↗

Explain why we cannot find inverse functions for all polynomial functions.

Answer

It can be too difficult or impossible to solve for

x

in terms of

y .

5.7-3 §5.7 source ↗

When finding the inverse of a radical function, what restriction will we need to make?

Answer

We will need a restriction on the domain of the answer.

5.7-5 §5.7 source ↗

For the following exercises, find the inverse of the function on the given domain.

$f (x) =$ ${(x - 4)}^{2},$ $[4, \infty)$

Answer

f^{- 1} (x) = \sqrt{x} + 4

5.7-7 §5.7 source ↗

For the following exercises, find the inverse of the function on the given domain.

$f (x) =$ ${(x + 1)}^{2} -$ $3,$ $[−1, \infty)$

Answer

f^{- 1} (x) =

\sqrt{x + 3} -

1

5.7-9 §5.7 source ↗

For the following exercises, find the inverse of the function on the given domain.

$f (x) =$ $12 -$ $x^{2},$ $[0, \infty)$

Answer

f^{- 1} (x) = \sqrt{12 - x}

5.7-11 §5.7 source ↗

For the following exercises, find the inverse of the function on the given domain.

$f (x) =$ $2 x^{2} +$ $4,$ $[0, \infty)$

Answer

f^{- 1} (x) = \sqrt{\frac{x - 4}{2}}

5.7-13 §5.7 source ↗

For the following exercises, find the inverse of the functions.

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

Answer

f^{- 1} (x) =

\sqrt{\frac{x - 1}{3} 3}

5.7-15 §5.7 source ↗

For the following exercises, find the inverse of the functions.

$f (x) = 4 - 2 x^{3}$

Answer

f^{- 1} (x) =

\sqrt{\frac{4 - x}{2} 3}

5.7-17 §5.7 source ↗

For the following exercises, find the inverse of the functions.

$f (x) = \sqrt{3 - 4 x}$

Answer

f^{−1} (x) =

\frac{3 - x^{2}}{4},

[0, \infty)

5.7-19 §5.7 source ↗

For the following exercises, find the inverse of the functions.

$f (x) = \sqrt{6 x - 8} + 5$

Answer

f^{−1} (x) =

\frac{{(x - 5)}^{2} + 8}{6}

5.7-21 §5.7 source ↗

For the following exercises, find the inverse of the functions.

$f (x) = 3 - \sqrt[3]{x}$

Answer

f^{−1} (x) =

{(3 - x)}^{2}

5.7-23 §5.7 source ↗

For the following exercises, find the inverse of the functions.

$f (x) = \frac{3}{x - 4}$

Answer

f^{−1} (x) =

\frac{4 x + 3}{x}

5.7-25 §5.7 source ↗

For the following exercises, find the inverse of the functions.

$f (x) = \frac{x - 2}{x + 7}$

Answer

f^{−1} (x) =

\frac{7 x - 3}{1 - x}

5.7-27 §5.7 source ↗

For the following exercises, find the inverse of the functions.

$f (x) =$ $\frac{5 x + 1}{2 - 5 x}$

Answer

f^{−1} (x) =

\frac{2 x - 1}{5 x + 5}

5.7-29 §5.7 source ↗

For the following exercises, find the inverse of the functions.

$f (x) =$ $x^{2} +$ $4 x +$ $1,$ $[−2, \infty)$

Answer

f^{−1} (x) =

\sqrt{x + 3} -

2

5.7-31 §5.7 source ↗

For the following exercises, find the inverse of the function and graph both the function and its inverse.

$f (x) =$ $x^{2} +$ $2,$ $x \geq 0$

Answer

A graph displays two curved lines on a coordinate plane with x and y axes ranging from -2 to 8. A blue curve starts near (0.5, 2) and curves upwards and to the right, appearing to be an exponential function. An orange curve starts near (2, 0.5) and curves to the right and slightly upwards, appearing to be a logarithmic function. The curves suggest an inverse relationship.

$f^{- 1} (x) = \sqrt{x - 2}$

5.7-33 §5.7 source ↗

For the following exercises, find the inverse of the function and graph both the function and its inverse.

$f (x) =$ ${(x + 3)}^{2},$ $x \geq -$ $3$

Answer

Two inverse functions plotted on a coordinate plane: an exponential curve (orange) in the second and first quadrants, and a logarithmic curve (blue) in the first and fourth quadrants.

$f^{- 1} (x) = \sqrt{x - 3}$

5.7-35 §5.7 source ↗

For the following exercises, find the inverse of the function and graph both the function and its inverse.

$f (x) = x^{3} + 3$

Answer

A graph displaying a blue curve and an orange curve on an x-y coordinate plane. The blue curve passes through (0,2) and (1,6), while the orange curve passes through (2,0) and (6,1). These curves are inverses of each other, reflected across the y=x line.

$f^{- 1} (x) = \sqrt[3]{x - 3}$

5.7-37 §5.7 source ↗

For the following exercises, find the inverse of the function and graph both the function and its inverse.

$f (x) =$ $x^{2} +$ $4 x,$ $x \geq -$ $2$

Answer

A Cartesian graph displays two distinct increasing functions: a blue curve with an upward concave shape and an orange curve with a downward concave shape.

$f^{- 1} (x) =$ $\sqrt{x + 4} - 2$

5.7-39 §5.7 source ↗

For the following exercises, find the inverse of the function and graph both the function and its inverse.

$f (x) = \frac{2}{x}$

Answer

The graph of the reciprocal function f(x)=1/x, showing two branches in the first and third quadrants. There is a vertical asymptote at x=0 (the y-axis) and a horizontal asymptote at y=0 (the x-axis).

5.7-41 §5.7 source ↗

For the following exercises, use a graph to help determine the domain of the functions.

$f (x) =$ $\sqrt{\frac{(x + 1) (x - 1)}{x}}$

Answer

A Cartesian coordinate system displays a graph with two separate curves. The first curve starts at approximately (-1, 0) and extends upwards as it approaches x=0 from the left, showing a steep increase. The second curve starts at approximately (1, 0) and extends towards the upper right, gradually increasing as x increases. The x-axis is labeled from -4 to 4, and the y-axis from -4 to 4, with grid lines at integer values.

$[- 1, 0) \cup [1, \infty)$

5.7-43 §5.7 source ↗

For the following exercises, use a graph to help determine the domain of the functions.

$f (x) =$ $\sqrt{\frac{x (x + 3)}{x - 4}}$

Answer

A graph displays a curve with a vertical asymptote at x=4. The curve descends from high y-values for x greater than 4, then gradually flattens out to the right, approaching a horizontal asymptote.

$[- 3, 0] \cup (4, \infty)$

5.7-45 §5.7 source ↗

For the following exercises, use a graph to help determine the domain of the functions.

$f (x) =$ $\sqrt{\frac{9 - x^{2}}{x + 4}}$

Answer

The graph displays a function f(x) on a Cartesian coordinate system. The x-axis and y-axis both range from -32 to 32, with grid lines every 8 units. The function exhibits a vertical asymptote near x = -4. From the left of this asymptote, the function approaches a value around y=6, then curves sharply downwards towards negative infinity as x decreases. From the right of x = -4, the function shoots upwards towards positive infinity. There appears to be another vertical asymptote near x = 1. As x approaches 1 from the left, the function goes towards negative infinity, while from the right, it rises towards positive infinity. Between these two asymptotes, the function forms a small curve with a local maximum around (0, 1). As x approaches negative infinity, the function seems to level off, approaching a horizontal asymptote around y = 8.

$[- ∞, - 4] \cdot [- 3, 3]$

5.7-47 §5.7 source ↗

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given.

$f (x) =$ $x^{3} +$ $x -$ $2,$ $y =$ $0,$ $1,$ $2$

Answer

A coordinate plane shows the graph of a cubic function. The curve passes through the origin (0,0) and extends from the third quadrant, through (1,1), into the first quadrant, showing increasing values as x increases.

$(- 2, 0),$ $(0, 1),$ $(8, 2)$

5.7-49 §5.7 source ↗

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given.

$f (x) =$ $x^{3} +$ $8 x -$ $4,$ $y =$ $-$ $1,$ $0,$ $1$

Answer

A graph showing a cubic function plotted on a coordinate plane. The blue curve passes through the origin (0,0) and increases from approximately (-2, -20) to (2, 20).

$(- 13, - 1),$ $(- 4, 0),$ $(5, 1)$

5.7-51 §5.7 source ↗

For the following exercises, find the inverse of the functions with $a, b, c$ positive real numbers.

$f (x) = a x^{3} + b$

Answer

f^{- 1} (x) =

\sqrt[3]{\frac{x - b}{a}}

5.7-53 §5.7 source ↗

For the following exercises, find the inverse of the functions with $a, b, c$ positive real numbers.

$f (x) = \sqrt{a x^{2} - b}$

Answer

f^{- 1} (x) =

\frac{\sqrt{x^{2} - b}}{a}

5.7-55 §5.7 source ↗

For the following exercises, find the inverse of the functions with $a, b, c$ positive real numbers.

$f (x) =$ $\frac{a x + b}{x + c}$

Answer

f^{- 1} (x) =

\frac{c x - b}{a - x}

5.7-57 §5.7 source ↗

For the following exercises, determine the function described and then use it to answer the question.

An object dropped from a height of 600 feet has a height, $h (t),$ in feet after $t$ seconds have elapsed, such that $h (t) =$ $600 -$ $16 t^{2} .$ Express $t$ as a function of height $h,$ and find the time to reach a height of 400 feet.

Answer

t (h) =

\sqrt{\frac{600 - h}{16}}

, 3.54 seconds

5.7-59 §5.7 source ↗

For the following exercises, determine the function described and then use it to answer the question.

The surface area, $A,$ of a sphere in terms of its radius, $r,$ is given by $A (r) = 4 π r^{2} .$ Express $r$ as a function of $A,$ and find the radius of a sphere with a surface area of 1000 square inches.

Answer

r (A) = \sqrt{\frac{A}{4 π}}, \approx

8.92 in.

5.7-61 §5.7 source ↗

For the following exercises, determine the function described and then use it to answer the question.

The period $T,$ in seconds, of a simple pendulum as a function of its length $l,$ in feet, is given by $T (l) =$ $2 π \sqrt{\frac{l}{32.2}}$ . Express $l$ as a function of $T$ and determine the length of a pendulum with period of 2 seconds.

Answer

l (T) =

32.2 (\frac{T}{2 π}), \approx

3.26 ft

5.7-63 §5.7 source ↗

For the following exercises, determine the function described and then use it to answer the question.

The surface area, $A,$ of a cylinder in terms of its radius, $r,$ and height, $h,$ is given by $A =$ $2 π r^{2} +$ $2 π r h .$ If the height of the cylinder is 4 feet, express the radius as a function of $A$ and find the radius if the surface area is 200 square feet.

Answer

r (A) = \sqrt{\frac{A + 8 π}{2 π}}

–2, 3.99 ft

5.7-65 §5.7 source ↗

For the following exercises, determine the function described and then use it to answer the question.

Consider a cone with height of 30 feet. Express the radius, $r,$ in terms of the volume, $V,$ and find the radius of a cone with volume of 1000 cubic feet.

Answer

r (V) = \sqrt{\frac{V}{10 π}},

≈ 5.64 ft

Try It #1 §5.8 source ↗

The quantity

y

varies directly with the square of

x .

y = 24

when

x = 3,

find

y

when

x

is 4.

Answer

\frac{128}{3}

Try It #2 §5.8 source ↗

A quantity

y

varies inversely with the square of

x .

y = 8

when

x = 3,

find

y

when

x

is 4.

Answer

\frac{9}{2}

Try It #3 §5.8 source ↗

A quantity

x

varies directly with the square of

y

and inversely with

z .

x = 40

when

y = 4

and

z = 2,

find

x

when

y = 10

and

z = 25.

Answer

x = 20

5.8-1 §5.8 source ↗

What is true of the appearance of graphs that reflect a direct variation between two variables?

Answer

The graph will have the appearance of a power function.

5.8-3 §5.8 source ↗

Is there a limit to the number of variables that can vary jointly? Explain.

Answer

No. Multiple variables may jointly vary.

5.8-5 §5.8 source ↗