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Chapter 3: Polynomial and Rational FunctionsContent check — Precalculus, Chapter 3: Polynomial and Rational Functions

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

Chapter 3: Polynomial and Rational Functions

432 problems · 80 figures

Try It #1 §3.1 source ↗

Express

\sqrt{- 24}

in standard form.

Answer

\sqrt{- 24} = 0 + 2 i \sqrt{6}

Try It #2 §3.1 source ↗

Plot the complex number

−4 - i

on the complex plane.

Answer

Graph of the plotted point, -4-i.

Try It #3 §3.1 source ↗

Subtract

2 + 5 i

from

3 - 4 i .

Answer

(3 - 4 i) -

(2 + 5 i) =

1 -

9 i

Try It #4 §3.1 source ↗

Find the product

- 4 (2 + 6 i) .

Answer

- 8 - 24 i

Try It #5 §3.1 source ↗

Multiply

(3 - 4 i) (2 + 3 i) .

Answer

18 + i

Try It #6 §3.1 source ↗

Let

f (x) =

2 x^{2} -

3 x .

Evaluate

f (8 - i) .

Answer

102 - 29 i

Try It #7 §3.1 source ↗

Let

f (x) =

\frac{x + 1}{x - 4} .

Evaluate

f (- i) .

Answer

- \frac{3}{17} + \frac{5 i}{17}

3.1-1 §3.1 source ↗

Explain how to add complex numbers.

Answer

Add the real parts together and the imaginary parts together.

3.1-3 §3.1 source ↗

Give an example to show the product of two imaginary numbers is not always imaginary.

Answer

i

times

i

equals –1, which is not imaginary. (answers vary)

3.1-5 §3.1 source ↗

For the following exercises, evaluate the algebraic expressions.

$If f (x) =$ $x^{2} +$ $x -$ $4,$ evaluate $f (2 i) .$

Answer

- 8 + 2 i

3.1-7 §3.1 source ↗

For the following exercises, evaluate the algebraic expressions.

$If f (x) =$ $x^{2} +$ $3 x +$ $5,$ evaluate $f (2 + i) .$

Answer

14 + 7 i

3.1-9 §3.1 source ↗

For the following exercises, evaluate the algebraic expressions.

$If f (x) =$ $\frac{x + 1}{2 - x},$ evaluate $f (5 i) .$

Answer

- \frac{23}{29} + \frac{15}{29} i

3.1-11 §3.1 source ↗

For the following exercises, determine the number of real and nonreal solutions for each quadratic function shown.

Graph of a parabola intersecting the real axis.

Answer

2 real and 0 nonreal

3.1-13 §3.1 source ↗

For the following exercises, plot the complex numbers on the complex plane.

$1 - 2 i$

Answer

Graph of the plotted point, 1-2i.

3.1-15 §3.1 source ↗

For the following exercises, plot the complex numbers on the complex plane.

$i$

Answer

Graph of the plotted point, i.

3.1-17 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$(3 + 2 i) +$ $(5 - 3 i)$

Answer

8 - i

3.1-19 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$(- 5 + 3 i) -$ $(6 - i)$

Answer

- 11 + 4 i

3.1-21 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$(- 4 + 4 i) -$ $(- 6 + 9 i)$

Answer

2 - 5 i

3.1-23 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$(5 - 2 i) (3 i)$

Answer

6 + 15 i

3.1-25 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$(- 2 + 4 i) (8)$

Answer

- 16 + 32 i

3.1-27 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$(- 1 + 2 i) (- 2 + 3 i)$

Answer

- 4 - 7 i

3.1-29 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$(3 + 4 i) (3 - 4 i)$

Answer

25

3.1-31 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\frac{6 - 2 i}{3}$

Answer

2 - \frac{2}{3} i

3.1-33 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\frac{6 + 4 i}{i}$

Answer

4 - 6 i

3.1-35 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\frac{3 + 4 i}{2 - i}$

Answer

\frac{2}{5} + \frac{11}{5} i

3.1-37 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\sqrt{- 9} + 3 \sqrt{- 16}$

Answer

15 i

3.1-39 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$\frac{2 + \sqrt{- 12}}{2}$

Answer

1 + i \sqrt{3}

3.1-41 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$i^{8}$

Answer

1

3.1-43 §3.1 source ↗

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$i^{22}$

Answer

- 1

3.1-45 §3.1 source ↗

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

Evaluate ${(1 - i)}^{k}$ for $k =$ $2, 6, and 10 .$ Predict the value if $k = 14.$

Answer

128i

3.1-47 §3.1 source ↗

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

Show that a solution of $x^{6} + 1 = 0$ is $\frac{\sqrt{3}}{2} + \frac{1}{2} i .$

Answer

{(\frac{\sqrt{3}}{2} + \frac{1}{2} i)}^{6} =

-

1

3.1-49 §3.1 source ↗

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

$\frac{1}{i} + \frac{4}{i^{3}}$

Answer

3 i

3.1-51 §3.1 source ↗

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

$i^{7} (1 + i^{2})$

Answer

0

3.1-53 §3.1 source ↗

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

$\frac{(2 + i) (4 - 2 i)}{(1 + i)}$

Answer

5 – 5i

3.1-55 §3.1 source ↗

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

$\frac{{(3 + i)}^{2}}{{(1 + 2 i)}^{2}}$

Answer

- 2 i

3.1-57 §3.1 source ↗

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

$\frac{4 + i}{i} +$ $\frac{3 - 4 i}{1 - i}$

Answer

\frac{9}{2} - \frac{9}{2} i

Try It #1 §3.2 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),

so

(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 §3.2 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 §3.2 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},

or

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

Try It #4 §3.2 source ↗

In a separate 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 §3.2 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

3.2-1 §3.2 source ↗

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

Answer

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

3.2-3 §3.2 source ↗

Explain why the condition of

a \neq 0

is imposed in the definition of the quadratic function.

Answer

If

a = 0

then the function becomes a linear function.

3.2-5 §3.2 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.

3.2-7 §3.2 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)

3.2-9 §3.2 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})

3.2-11 §3.2 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)

3.2-13 §3.2 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})

3.2-15 §3.2 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} .

3.2-17 §3.2 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} .

3.2-19 §3.2 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.

3.2-21 §3.2 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) .

3.2-23 §3.2 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) .

3.2-25 §3.2 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) .

3.2-27 §3.2 source ↗

For the following exercises, solve the equations over the complex numbers.

$x^{2} = - 8$

Answer

{2 i \sqrt{2}, - 2 i \sqrt{2}}

3.2-29 §3.2 source ↗

For the following exercises, solve the equations over the complex numbers.

$x^{2} + 27 = 0$

Answer

{3 i \sqrt{3}, - 3 i \sqrt{3}}

3.2-31 §3.2 source ↗

For the following exercises, solve the equations over the complex numbers.

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

Answer

{2 + i, 2 - i}

3.2-33 §3.2 source ↗

For the following exercises, solve the equations over the complex numbers.

$x^{2} - 4 x + 13 = 0$

Answer

{2 + 3 i, 2 - 3 i}

3.2-35 §3.2 source ↗

For the following exercises, solve the equations over the complex numbers.

$x^{2} - 10 x + 26 = 0$

Answer

{5 + i, 5 - i}

3.2-37 §3.2 source ↗

For the following exercises, solve the equations over the complex numbers.

$x (x - 4) = 20$

Answer

{2 + 2 \sqrt{6}, 2 - 2 \sqrt{6}}

3.2-39 §3.2 source ↗

For the following exercises, solve the equations over the complex numbers.

$2 x^{2} + 2 x + 5 = 0$

Answer

{- \frac{1}{2} + \frac{3}{2} i, - \frac{1}{2} - \frac{3}{2} i}

3.2-41 §3.2 source ↗

For the following exercises, solve the equations over the complex numbers.

$5 x^{2} + 6 x + 2 = 0$

Answer

{- \frac{3}{5} + \frac{1}{5} i, - \frac{3}{5} - \frac{1}{5} i}

3.2-43 §3.2 source ↗

For the following exercises, solve the equations over the complex numbers.

$x^{2} + x + 2 = 0$

Answer

{- \frac{1}{2} + \frac{1}{2} i \sqrt{7}, - \frac{1}{2} - \frac{1}{2} i \sqrt{7}}

3.2-45 §3.2 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.

$(h, k) =$ $(2, 0),$ $(x, y) =$ $(4, 4)$

Answer

f (x) =

x^{2} -

4 x +

4

3.2-47 §3.2 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.

$(h, k) =$ $(0, 1),$ $(x, y) =$ $(2, 5)$

Answer

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

3.2-49 §3.2 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.

$(h, k) =$ $(- 5, 3),$ $(x, y) =$ $(2, 9)$

Answer

f (x) =

\frac{6}{49} x^{2} +

\frac{60}{49} x +

\frac{297}{49}

3.2-51 §3.2 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.

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

Answer

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

3.2-53 §3.2 source ↗

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

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

Answer

Graph of f(x) = x^2-2x

Vertex $(1, - 1),$ Axis of symmetry is $x = 1.$ Intercepts are $(0, 0),$ $(2, 0) .$

3.2-55 §3.2 source ↗

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

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

Answer

Graph of f(x)x^2-5x-6

Vertex $(\frac{5}{2}, \frac{- 49}{4}),$ Axis of symmetry is $x = \frac{5}{2},$ intercepts: $(6, 0),$ $(- 1, 0) .$

3.2-57 §3.2 source ↗

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

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

Answer

Graph of f(x)=-2x^2+5x-8

Vertex $(\frac{5}{4}, - \frac{39}{8}),$ Axis of symmetry is $x = \frac{5}{4} .$ Intercepts are $(0, - 8) .$

3.2-59 §3.2 source ↗

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

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

Answer

f (x) =

x^{2} -

4 x +

1

3.2-61 §3.2 source ↗

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

Graph of a negative parabola with a vertex at (2, 7).

Answer

f (x) =

-

2 x^{2} +

8 x -

1

3.2-63 §3.2 source ↗

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

Graph of a positive parabola with a vertex at (3, -1) and y-intercept at (0, 3.5).

Answer

f (x) =

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

3 x +

\frac{7}{2}

3.2-65 §3.2 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
$y$	5	2	1	2	5

Answer

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

3.2-67 §3.2 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
$y$	–2	1	2	1	–2

Answer

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

3.2-69 §3.2 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
$y$	8	2	0	2	8

Answer

f (x) = 2 x^{2}

3.2-71 §3.2 source ↗

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

Graph on the same set of axes $f (x) =$ $x^{2},$ $f (x) =$ $x^{2} +$ $2$ and $f (x) =$ $x^{2},$ $f (x) =$ $x^{2} +$ $5$ and $f (x) = x^{2} - 3.$ What appears to be the effect of adding a constant?

Answer

The graph is shifted up or down (a vertical shift).

3.2-73 §3.2 source ↗

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

The path of an object projected at a 45 degree angle with initial velocity of 80 feet per second is given by the function $h (x) =$ $\frac{- 32}{{(80)}^{2}} x^{2} +$ $x$ where $x$ is the horizontal distance traveled and $h (x)$ is the height in feet. Use the TRACE feature of your calculator to determine the height of the object when it has traveled 100 feet away horizontally.

Answer

50 feet

3.2-75 §3.2 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 up.

Answer

Domain is

(- \infty, \infty) .

Range is

[- 2, \infty) .

3.2-77 §3.2 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 $(−5, 11),$ opens down.

Answer

Domain is

(- \infty, \infty)

Range is

(- \infty, 11] .

3.2-79 §3.2 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 $(1, 1)$ and has shape of $f (x) = 2 x^{2} .$ Vertex is on the $y -$ axis.

Answer

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

3.2-81 §3.2 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 $(2, 3)$ and has the shape of $f (x) = 3 x^{2} .$ Vertex is on the $y -$ axis.

Answer

f (x) = 3 x^{2} - 9

3.2-83 §3.2 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 $(4, 3)$ and has the shape of $f (x) = 5 x^{2} .$ Vertex is on the $y -$ axis.

Answer

f (x) = 5 x^{2} - 77

3.2-85 §3.2 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 corral producing the greatest enclosed area given 200 feet of fencing.

Answer

50 feet by 50 feet. Maximize

f (x) =

-

x^{2} +

100 x .

3.2-87 §3.2 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 corral producing the greatest enclosed area split into 3 pens of the same size given 500 feet of fencing.

Answer

125 feet by 62.5 feet. Maximize

f (x) =

-

2 x^{2} +

250 x .

3.2-89 §3.2 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 difference is 12, find the pair with the smallest product. What is the product?

Answer

6

and

- 6;

product is –36; maximize

f (x) =

x^{2} +

12 x .

3.2-91 §3.2 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 rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by $h (t) =$ $-$ $4.9 t^{2} +$ $229 t +$ $234.$ Find the maximum height the rocket attains.

Answer

2909.56 meters

3.2-93 §3.2 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 soccer stadium holds 62,000 spectators. With a ticket price of $11, the average attendance has been 26,000. When the price dropped to $9, the average attendance rose to 31,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue?

Answer

$10.70

Try It #1 §3.3 source ↗

Which functions are power functions?

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

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 §3.3 source ↗

Describe in words and symbols the end behavior of

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

Answer

As

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 §3.3 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 §3.3 source ↗

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

Graph of an even-degree polynomial.

Figure 8

Answer

As

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 §3.3 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 §3.3 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 §3.3 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 §3.3 source ↗

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

Graph of an odd-degree polynomial.

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 §3.3 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.

3.3-1 §3.3 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.

3.3-3 §3.3 source ↗

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

Answer

As

x

decreases without bound, so does

f (x) .

As

x

increases without bound, so does

f (x) .

3.3-5 §3.3 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.

3.3-7 §3.3 source ↗

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

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

Answer

f (x)

is a power function because it contains a variable base raised to a fixed power. It is also a polynomial, with all coefficients except one equal to zero.

3.3-9 §3.3 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

3.3-11 §3.3 source ↗

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

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

Answer

Neither

3.3-13 §3.3 source ↗

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

$7 - 2 x^{2}$

Answer

Degree = 2, Coefficient = –2

3.3-15 §3.3 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

3.3-17 §3.3 source ↗

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

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

Answer

As

x \to \infty

,

f (x) \to \infty

, as

x \to - \infty

,

f (x) \to \infty

3.3-19 §3.3 source ↗

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

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

Answer

As

x \to - \infty

,

f (x) \to - \infty

, as

x \to \infty

,

f (x) \to - \infty

3.3-21 §3.3 source ↗

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

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

Answer

As

x \to - \infty

,

f (x) \to - \infty

,as

x \to \infty

,

f (x) \to - \infty

3.3-23 §3.3 source ↗

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

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

Answer

As

x \to \infty

,

f (x) \to \infty

, as

x \to - \infty

,

f (x) \to - \infty

3.3-25 §3.3 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) .

3.3-27 §3.3 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) .

3.3-29 §3.3 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) .

3.3-31 §3.3 source ↗

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

Graph of an odd-degree polynomial.

Answer

3

3.3-33 §3.3 source ↗

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

Graph of an odd-degree polynomial.

Answer

5

3.3-35 §3.3 source ↗

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

Graph of an odd-degree polynomial.

Answer

3

3.3-37 §3.3 source ↗

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

Graph of an odd-degree polynomial.

Answer

5

3.3-39 §3.3 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.

Graph of an odd-degree polynomial.

Answer

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

3.3-41 §3.3 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.

Graph of an even-degree polynomial.

Answer

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

3.3-43 §3.3 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.

Graph of an odd-degree polynomial.

Answer

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

3.3-44 §3.3 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.

Graph of an equation.

Answer

No.

3.3-45 §3.3 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.

Graph of an odd-degree polynomial.

Answer

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

3.3-47 §3.3 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 $x \to \infty$ , $f (x) \to \infty$

3.3-49 §3.3 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 $x \to \infty$ , $f (x) \to - \infty$

3.3-51 §3.3 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

Graph of f(x)=x^3(x-2).

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

3.3-53 §3.3 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

Graph of f(x)=x(14-2x)(10-2x).

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 $x \to \infty$ , $f (x) \to \infty$

3.3-55 §3.3 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 of a cubic function is displayed on a Cartesian coordinate system. The x-axis ranges from -10 to 10, and the y-axis ranges from -400 to 300. The blue curve represents the function, exhibiting an S-shape characteristic of a cubic polynomial. It passes through the origin (0,0) and has approximate x-intercepts at x = -4 and x = 2. The function increases, reaches a local maximum near x = -2, then decreases to a local minimum near x = 2, and subsequently increases again.

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

3.3-57 §3.3 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

Graph of f(x)=x^3-27.

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

3.3-59 §3.3 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

Graph of f(x)=-x^3+x^2+2x.

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 $x \to \infty$ , $f (x) \to \infty$

3.3-61 §3.3 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

3.3-63 §3.3 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 $(0, 0) .$ 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

3.3-65 §3.3 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 $(0, 1) .$ 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

3.3-67 §3.3 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

3.3-69 §3.3 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 §3.4 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 §3.4 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 2.

Try It #3 §3.4 source ↗

Sketch a graph of

f (x) =

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

Answer

Graph of f(x)=(1/4)x(x-1)^4(x+3)^3.

Try It #4 §3.4 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 §3.4 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 §3.4 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) .

3.4-1 §3.4 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.

3.4-3 §3.4 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 .

3.4-5 §3.4 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.

3.4-7 §3.4 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)

3.4-9 §3.4 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)

3.4-11 §3.4 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)

3.4-13 §3.4 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)

3.4-15 §3.4 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)

3.4-17 §3.4 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)

3.4-19 §3.4 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)

3.4-21 §3.4 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)

3.4-23 §3.4 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)

3.4-25 §3.4 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.

3.4-27 §3.4 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.

3.4-29 §3.4 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.

3.4-31 §3.4 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

3.4-33 §3.4 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

3.4-35 §3.4 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

3.4-37 §3.4 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

3.4-39 §3.4 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

3.4-41 §3.4 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

3.4-43 §3.4 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 $x \to - \infty$ , $g (x) \to - \infty$ , as $x \to \infty$ , $g (x) \to \infty$ .

Graph of g(x)=(x+4)(x-1)^2.

3.4-45 §3.4 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 $x \to - \infty$ , $k (x) \to - \infty$ , as $x \to \infty$ , $k (x) \to \infty .$

Graph of k(x)=(x-3)^3(x-2)^2.

3.4-47 §3.4 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 $x \to - \infty$ , $n (x) \to \infty$ , as $x \to \infty$ , $n (x) \to - \infty .$

Graph of n(x)=-3x(x+2)(x-4).

3.4-49 §3.4 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)

3.4-51 §3.4 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)

3.4-53 §3.4 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

3.4-55 §3.4 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

3.4-57 §3.4 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)

3.4-59 §3.4 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)

3.4-61 §3.4 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}

3.4-63 §3.4 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)

3.4-65 §3.4 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)

3.4-67 §3.4 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)

3.4-69 §3.4 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)

3.4-71 §3.4 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)

3.4-73 §3.4 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)

3.4-75 §3.4 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

3.4-77 §3.4 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

3.4-79 §3.4 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 §3.5 source ↗

Divide

16 x^{3} -

12 x^{2} +

20 x -

3

by

4 x + 5.

Answer

4 x^{2} -

8 x +

15 -

\frac{78}{4 x + 5}

Try It #2 §3.5 source ↗

Use synthetic division to divide

3 x^{4} +

18 x^{3} -

3 x +

40

by

x + 7.

Answer

3 x^{3} -

3 x^{2} +

21 x -

150 +

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

Try It #3 §3.5 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

3.5-1 §3.5 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.

3.5-3 §3.5 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

3.5-5 §3.5 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

3.5-7 §3.5 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

3.5-9 §3.5 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

3.5-11 §3.5 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

3.5-13 §3.5 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

3.5-15 §3.5 source ↗

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

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

Answer

2 x^{2} +

2 x +

1 +

\frac{10}{x - 4}

3.5-17 §3.5 source ↗

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

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

Answer

2 x^{2} -

7 x +

1 -

\frac{2}{2 x + 1}

3.5-19 §3.5 source ↗

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

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

Answer

3 x^{2} -

11 x +

34 -

\frac{106}{x + 3}

3.5-21 §3.5 source ↗

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

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

Answer

x^{2} + 5 x + 1

3.5-23 §3.5 source ↗

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

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

Answer

4 x^{2} -

21 x +

84 -

\frac{323}{x + 4}

3.5-25 §3.5 source ↗

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

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

Answer

x^{2} - 14 x + 49

3.5-27 §3.5 source ↗

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

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

Answer

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

3.5-29 §3.5 source ↗

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

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

Answer

x^{3} - 3 x + 1

3.5-31 §3.5 source ↗

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

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

Answer

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

3.5-33 §3.5 source ↗

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

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

Answer

x^{3} -

6 x^{2} +

12 x -

8

3.5-35 §3.5 source ↗

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

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

Answer

x^{3} -

9 x^{2} +

27 x -

27

3.5-37 §3.5 source ↗

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

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

Answer

2 x^{3} - 2 x + 2

3.5-39 §3.5 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)

3.5-41 §3.5 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)

3.5-43 §3.5 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

No

3.5-45 §3.5 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)

3.5-47 §3.5 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)

3.5-49 §3.5 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

3.5-51 §3.5 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

3.5-53 §3.5 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

3.5-55 §3.5 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

3.5-57 §3.5 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}

3.5-59 §3.5 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}

3.5-61 §3.5 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}

3.5-63 §3.5 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}

3.5-65 §3.5 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

3.5-67 §3.5 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

3.5-69 §3.5 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

3.5-71 §3.5 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

3.5-73 §3.5 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 §3.6 source ↗

Use the Remainder Theorem to evaluate

f (x) =

2 x^{5} -

3 x^{4} -

9 x^{3} +

8 x^{2} +

2

at

x = - 3.

Answer

f (- 3) = - 412

Try It #2 §3.6 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 §3.6 source ↗

Use the Rational Zero Theorem to show that there are no rational zeros of

f (x) =

x^{3} -

5 x^{2} +

2 x +

1.

Answer

There are no rational zeros.

Try It #4 §3.6 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 §3.6 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 §3.6 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 §3.6 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

3.6-1 §3.6 source ↗

Describe a use for the Remainder Theorem.

Answer

The theorem can be used to evaluate a polynomial.

3.6-3 §3.6 source ↗

What is the difference between rational and real zeros?

Answer

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

3.6-5 §3.6 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.

3.6-7 §3.6 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

3.6-9 §3.6 source ↗

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

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

Answer

0

3.6-11 §3.6 source ↗

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

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

Answer

255

3.6-13 §3.6 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

3.6-15 §3.6 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}

3.6-17 §3.6 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

3.6-19 §3.6 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

3.6-21 §3.6 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}

3.6-23 §3.6 source ↗

For the following exercises, use the Rational Zero Theorem to find all real zeros.

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

Answer

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

3.6-25 §3.6 source ↗

For the following exercises, use the Rational Zero Theorem to find all real zeros.

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

Answer

4, - 4, - 5

3.6-27 §3.6 source ↗

For the following exercises, use the Rational Zero Theorem to find all real zeros.

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

Answer

5, - 3, - \frac{1}{2}

3.6-29 §3.6 source ↗

For the following exercises, use the Rational Zero Theorem to find all real zeros.

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

Answer

\frac{1}{2},

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

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

3.6-31 §3.6 source ↗

For the following exercises, use the Rational Zero Theorem to find all real zeros.

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

Answer

\frac{3}{2}

3.6-33 §3.6 source ↗

For the following exercises, use the Rational Zero Theorem to find all real zeros.

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

Answer

2, 3, - 1, - 2

3.6-35 §3.6 source ↗

For the following exercises, use the Rational Zero Theorem to find all real zeros.

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

Answer

\frac{1}{2}, - \frac{1}{2}, 2, - 3

3.6-37 §3.6 source ↗

For the following exercises, use the Rational Zero Theorem to find all real zeros.

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

Answer

- 1, - 1, \sqrt{5}, - \sqrt{5}

3.6-39 §3.6 source ↗

For the following exercises, use the Rational Zero Theorem to find all real zeros.

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

Answer

- \frac{3}{4}, - \frac{1}{2}

3.6-41 §3.6 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

3.6-43 §3.6 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

3.6-45 §3.6 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

3.6-47 §3.6 source ↗

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination.

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

Answer

1 positive, 1 negative

Graph of f(x)=x^4-x^2-1.

3.6-49 §3.6 source ↗

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination.

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

Answer

3 or 1 positive, 0 negative

Graph of f(x)=x^3-2x^2+x-1.

3.6-51 §3.6 source ↗

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination.

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

Answer

0 positive, 3 or 1 negative

Graph of f(x)=2x^3+37x^2+200x+300.

3.6-53 §3.6 source ↗

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination.

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

Answer

2 or 0 positive, 2 or 0 negative

Graph of f(x)=2x^4-5x^3-5x^2+5x+3.

3.6-55 §3.6 source ↗

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination.

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

Answer

2 or 0 positive, 2 or 0 negative

Graph of f(x)=10x^4-21x^2+11.

3.6-57 §3.6 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}

3.6-59 §3.6 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}

3.6-61 §3.6 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}

3.6-63 §3.6 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}

3.6-65 §3.6 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}

3.6-67 §3.6 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)

3.6-69 §3.6 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)

3.6-71 §3.6 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

3.6-73 §3.6 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

3.6-75 §3.6 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

3.6-77 §3.6 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

3.6-79 §3.6 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 §3.7 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 §3.7 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 §3.7 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 §3.7 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 §3.7 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 §3.7 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 §3.7 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 §3.7 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.

3.7-1 §3.7 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.

3.7-3 §3.7 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.

3.7-5 §3.7 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.

3.7-7 §3.7 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

3.7-9 §3.7 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

3.7-11 §3.7 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}

3.7-13 §3.7 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

3.7-15 §3.7 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

3.7-17 §3.7 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

3.7-19 §3.7 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} .

3.7-21 §3.7 source ↗

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

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

Answer

none

3.7-23 §3.7 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})

3.7-25 §3.7 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: $x \to$ $\pm \infty,$ $f (x) \to$ $\frac{1}{2}$

3.7-27 §3.7 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

3.7-29 §3.7 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: $x \to$ ${\frac{1}{3}}^{+},$ $f (x) \to$ $–∞,$ $x \to$ ${\frac{1}{3}}^{-},$ $f (x) \to$ $\infty,$ $x \to$ $-$ ${\frac{5}{2}}^{-},$ $f (x) \to$ $–∞,$ $x \to$ $-$ ${\frac{5}{2}}^{+}$ , $f (x) \to \infty$

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

3.7-31 §3.7 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

3.7-33 §3.7 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

3.7-35 §3.7 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

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

Graph of a rational function.

3.7-37 §3.7 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

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

Graph of a rational function.

3.7-39 §3.7 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.

3.7-41 §3.7 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.

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

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.

3.7-43 §3.7 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.

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

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.

3.7-45 §3.7 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.

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

Answer

$V . A . x =$ $-$ $1,$ $H . A . y =$ $1;$ $(- 3, 0);$ $(0, 3)$ ; removable discontinuity (hole) at $(1, 2)$

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

3.7-47 §3.7 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.

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

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.

3.7-49 §3.7 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.

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

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.

3.7-51 §3.7 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}

3.7-53 §3.7 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}

3.7-55 §3.7 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}

3.7-57 §3.7 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}

3.7-59 §3.7 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

27(x - 2) / ((x - 3)² (x + 3))

y =

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

3.7-61 §3.7 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}

3.7-63 §3.7 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}}

3.7-65 §3.7 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 $y = 0$

3.7-67 §3.7 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 $y = 2$

3.7-69 §3.7 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 $y = 1$

3.7-71 §3.7 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

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

Graph of f(x)=4/(2x-3).

3.7-73 §3.7 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

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

Graph of f(x)=(x+2)/(x-1)(x-4).

3.7-75 §3.7 source ↗

For the following exercises, identify the removable discontinuity.

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

Answer

(2, 4)

3.7-77 §3.7 source ↗

For the following exercises, identify the removable discontinuity.

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

Answer

(2, 5)

3.7-79 §3.7 source ↗

For the following exercises, identify the removable discontinuity.

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

Answer

(- 1, 1)

3.7-81 §3.7 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}

3.7-83 §3.7 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.

3.7-85 §3.7 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.

3.7-87 §3.7 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 §3.8 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 §3.8 source ↗

Find the inverse function of

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

Answer

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

Try It #3 §3.8 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 §3.8 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 §3.8 source ↗

Find the inverse of the function

f (x) =

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

Answer

f^{- 1} (x) =

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

3.8-1 §3.8 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 .

3.8-3 §3.8 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.

3.8-5 §3.8 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

3.8-7 §3.8 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

3.8-9 §3.8 source ↗

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

$f (x) =$ $3 x^{2} +$ $5,$ $(- \infty, 0]$

Answer

f^{- 1} (x) =

-

\sqrt{\frac{x - 5}{3}}

3.8-11 §3.8 source ↗

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

$f (x) =$ $9 -$ $x^{2},$ $[0, \infty)$

Answer

f (x) = \sqrt{9 - x}

3.8-13 §3.8 source ↗

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

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

Answer

f^{- 1} (x) = \sqrt[3]{x - 5}

3.8-15 §3.8 source ↗

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

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

Answer

f^{- 1} (x) = \sqrt[3]{4 - x}

3.8-17 §3.8 source ↗

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

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

Answer

f^{- 1} (x) =

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

[0, \infty)

3.8-19 §3.8 source ↗

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

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

Answer

f^{- 1} (x) =

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

[9, \infty)

3.8-21 §3.8 source ↗

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

$f (x) = 9 + 2 \sqrt[3]{x}$

Answer

f^{- 1} (x) =

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

3.8-23 §3.8 source ↗

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

$f (x) = \frac{2}{x + 8}$

Answer

f^{- 1} (x) =

{\frac{2 - 8 x}{x}}^{}

3.8-25 §3.8 source ↗

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

$f (x) = \frac{x + 3}{x + 7}$

Answer

f^{- 1} (x) =

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

3.8-27 §3.8 source ↗

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

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

Answer

f^{- 1} (x) =

\frac{5 x - 4}{4 x + 3}

3.8-29 §3.8 source ↗

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

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

Answer

f^{- 1} (x) =

\sqrt{x + 1} -

1

3.8-31 §3.8 source ↗

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

$f (x) =$ $x^{2} -$ $6 x +$ $3,$ $[3, \infty)$

Answer

f^{- 1} (x) =

\sqrt{x + 6} +

3

3.8-33 §3.8 source ↗

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

$f (x) =$ $4 -$ $x^{2},$ $x \geq 0$

Answer

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

Graph of f(x)=4- x^2 and its inverse, f^(-1)(x)= sqrt(4-x).

3.8-35 §3.8 source ↗

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

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

Answer

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

Graph of f(x)= (x-4)^2 and its inverse, f^(-1)(x)= sqrt(x)+4.

3.8-37 §3.8 source ↗

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

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

Answer

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

Graph of f(x)= 1-x^3 and its inverse, f^(-1)(x)= (1-x)^(1/3).

3.8-39 §3.8 source ↗

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

$f (x) =$ $x^{2} -$ $6 x +$ $1,$ $x \geq 3$

Answer

$f^{- 1} (x) =$ $\sqrt{x + 8} +$ $3$

Graph of f(x)= x^2-6x+1 and its inverse, f^(-1)(x)= sqrt(x+8)+3.

3.8-41 §3.8 source ↗

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

$f (x) =$ $\frac{1}{x^{2}},$ $x \geq 0$

Answer

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

Graph of f(x)= 1/x^2 and its inverse, f^(-1)(x)= sqrt(1/x).

3.8-43 §3.8 source ↗

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

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

Answer

$[- 2, 1) \cup [3, \infty)$

Graph of f(x)= sqrt((x+2)(x-3)/(x-1)).

3.8-45 §3.8 source ↗

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

$f (x) =$ $\sqrt{\frac{x^{2} - x - 20}{x - 2}}$

Answer

$[- 4, 2) \cup [5, \infty)$

Graph of f(x)= sqrt((x^2-x-20)/(x-2)).

3.8-47 §3.8 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.

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

Answer

$(- 2, 0);$ $(4, 2);$ $(22, 3)$

Graph of f(x)= x^3-x-2.

3.8-49 §3.8 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.

$f (x) =$ $x^{3} +$ $3 x -$ $4,$ $y =$ $0,$ $1,$ $2$

Answer

$(- 4, 0);$ $(0, 1);$ $(10, 2)$

Graph of f(x)= x^3+3x-4.

3.8-51 §3.8 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.

$f (x) =$ $x^{4} +$ $5 x +$ $1,$ $y =$ $-$ $1,$ $0,$ $1$

Answer

$(- 3, - 1);$ $(1, 0);$ $(7, 1)$

Graph of f(x)= x^4+5x+1.

3.8-53 §3.8 source ↗

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

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

Answer

f^{- 1} (x) =

\sqrt{x + \frac{b^{2}}{4}} -

\frac{b}{2}

3.8-55 §3.8 source ↗

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

$f (x) = \sqrt[3]{a x + b}$

Answer

f^{- 1} (x) =

\frac{x^{3} - b}{a}

3.8-57 §3.8 source ↗

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

An object dropped from a height of 200 meters has a height, $h (t),$ in meters after $t$ seconds have lapsed, such that $h (t) =$ $200 -$ $4.9 t^{2} .$ Express $t$ as a function of height, $h,$ and find the time to reach a height of 50 meters.

Answer

t (h) =

\sqrt{\frac{200 - h}{4.9}},

5.53 seconds

3.8-59 §3.8 source ↗

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

The volume, $V,$ of a sphere in terms of its radius, $r,$ is given by $V (r) = \frac{4}{3} π r^{3} .$ Express $r$ as a function of $V,$ and find the radius of a sphere with volume of 200 cubic feet.

Answer

r (V) = \sqrt[3]{\frac{3 V}{4 π}},

3.63 feet

3.8-61 §3.8 source ↗

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

A container holds 100 ml of a solution that is 25 ml acid. If $n$ ml of a solution that is 60% acid is added, the function $C (n) =$ $\frac{25 + .6 n}{100 + n}$ gives the concentration, $C,$ as a function of the number of ml added, $n .$ Express $n$ as a function of $C$ and determine the number of mL that need to be added to have a solution that is 50% acid.

Answer

n (C) =

\frac{100 C - 25}{0.6 - C},

250 mL

3.8-63 §3.8 source ↗

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

The volume of a cylinder , $V,$ in terms of radius, $r,$ and height, $h,$ is given by $V = π r^{2} h .$ If a cylinder has a height of 6 meters, express the radius as a function of $V$ and find the radius of a cylinder with volume of 300 cubic meters.

Answer

r (V) = \sqrt{\frac{V}{6 π}},

3.99 meters

3.8-65 §3.8 source ↗

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

The volume of a right circular cone, $V,$ in terms of its radius, $r,$ and its height, $h,$ is given by $V = \frac{1}{3} π r^{2} h .$ Express $r$ in terms of $V$ if the height of the cone is 12 inches and find the radius of a cone with volume of 50 cubic inches.

Answer

r (V) = \sqrt{\frac{V}{4 π}},

1.99 inches

Try It #1 §3.9 source ↗

The quantity

y

varies directly with the square of

x .

If

y = 24

when

x = 3,

find

y

when

x

is 4.

Answer

\frac{128}{3}

Try It #2 §3.9 source ↗

A quantity

y

varies inversely with the square of

x .

If

y = 8

when

x = 3,

find

y

when

x

is 4.

Answer

\frac{9}{2}

Try It #3 §3.9 source ↗

x

varies directly with the square of

y

and inversely with

z .

If

x = 40

when

y = 4

and

z = 2,

find

x

when

y = 10

and

z = 25.

Answer

x = 20

3.9-1 §3.9 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.

3.9-3 §3.9 source ↗

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

Answer

No. Multiple variables may jointly vary.

3.9-5 §3.9 source ↗

For the following exercises, write an equation describing the relationship of the given variables.

$y$ varies directly as the square of $x$ and when $x = 4, y = 80 .$

Answer

y = 5 x^{2}

3.9-7 §3.9 source ↗

For the following exercises, write an equation describing the relationship of the given variables.

$y$ varies directly as the cube of $x$ and when $x = 36, y = 24.$

Answer

y = \frac{1}{1944} x^{3}

3.9-9 §3.9 source ↗

For the following exercises, write an equation describing the relationship of the given variables.

$y$ varies directly as the fourth power of $x$ and when $x = 1, y = 6.$

Answer

y = 6 x^{4}

3.9-11 §3.9 source ↗

For the following exercises, write an equation describing the relationship of the given variables.

$y$ varies inversely as the square of $x$ and when $x = 3, y = 2.$

Answer

y = \frac{18}{x^{2}}

3.9-13 §3.9 source ↗

For the following exercises, write an equation describing the relationship of the given variables.

$y$ varies inversely as the fourth power of $x$ and when $x = 3, y = 1.$

Answer

y = \frac{81}{x^{4}}

3.9-15 §3.9 source ↗

For the following exercises, write an equation describing the relationship of the given variables.

$y$ varies inversely as the cube root of $x$ and when $x = 64, y = 5.$

Answer

y = \frac{20}{\sqrt[3]{x}}

3.9-17 §3.9 source ↗

For the following exercises, write an equation describing the relationship of the given variables.

$y$ varies jointly as $x$ , $z$ , and $w$ and when $x = 1$ , $z = 2$ , $w = 5$ , then $y = 100.$

Answer

y = 10 x z w

3.9-19 §3.9 source ↗

For the following exercises, write an equation describing the relationship of the given variables.

$y$ varies jointly as $x$ and the square root of $z$ and when $x = 2$ and $z = 25$ , then $y = 100.$

Answer

y = 10 x \sqrt{z}

3.9-21 §3.9 source ↗

For the following exercises, write an equation describing the relationship of the given variables.

$y$ varies jointly as $x$ and $z$ and inversely as $w .$ When $x = 3$ , $z = 5$ , and $w = 6$ , then $y = 10.$

Answer

y = 4 \frac{x z}{w}

3.9-23 §3.9 source ↗

For the following exercises, write an equation describing the relationship of the given variables.

$y$ varies jointly as $x$ and $z$ and inversely as the square root of $w$ and the square of $t .$ When $x = 3$ , $z = 1, w = 25$ , and $t = 2$ , then $y = 6.$

Answer

y = 40 \frac{x z}{\sqrt{w} t^{2}}

3.9-25 §3.9 source ↗

For the following exercises, use the given information to find the unknown value.

$y$ varies directly as the square of $x$ . When $x = 2$ , then $y = 16$ . Find $y$ when $x = 8$ .

Answer

y = 256

3.9-27 §3.9 source ↗

For the following exercises, use the given information to find the unknown value.

$y$ varies directly as the square root of $x .$ When $x = 16$ , then $y = 4$ . Find $y when x =$ $36$ .

Answer

y = 6

3.9-29 §3.9 source ↗

For the following exercises, use the given information to find the unknown value.

$y$ varies inversely with $x .$ When $x = 3$ , then $y = 2$ . Find $y$ when $x = 1$ .

Answer

y = 6

3.9-31 §3.9 source ↗

For the following exercises, use the given information to find the unknown value.

$y$ varies inversely with the cube of $x .$ When $x = 3$ , then $y = 1$ . Find $y$ when $x = 1$ .

Answer

y = 27

3.9-33 §3.9 source ↗

For the following exercises, use the given information to find the unknown value.

$y$ varies inversely with the cube root of $x .$ When $x = 27,$ then $y = 5.$ Find $y$ when $x = 125.$

Answer

y = 3

3.9-35 §3.9 source ↗

For the following exercises, use the given information to find the unknown value.

$y$ varies jointly as $x, z, and w .$ When $x = 2,$ $z = 1,$ and $w = 12,$ then $y = 72.$ Find $y$ when $x = 1,$ $z = 2,$ and $w = 3.$

Answer

y = 18

3.9-37 §3.9 source ↗

For the following exercises, use the given information to find the unknown value.

$y$ varies jointly as the square of $x$ and the square root of $z .$ When $x = 2$ and $z = 9,$ then $y = 24.$ Find $y$ when $x = 3$ and $z = 25.$

Answer

y = 90

3.9-39 §3.9 source ↗

For the following exercises, use the given information to find the unknown value.

$y$ varies jointly as the square of $x$ and the cube of $z$ and inversely as the square root of $w .$ When $x = 2,$ $z = 2,$ and $w = 64,$ then $y = 12.$ Find $y$ when $x = 1,$ $z = 3,$ and $w = 4.$

Answer

y = \frac{81}{2}

3.9-41 §3.9 source ↗

For the following exercises, use a calculator to graph the equation implied by the given variation.

$y$ varies directly with the square of $x$ and when $x = 2, y = 3.$

Answer

$y = \frac{3}{4} x^{2}$

Graph of y=3/4(x^2).

3.9-43 §3.9 source ↗

For the following exercises, use a calculator to graph the equation implied by the given variation.

$y$ varies directly as the square root of $x$ and when $x = 36, y = 2.$

Answer

$y = \frac{1}{3} \sqrt{x}$

Graph of y=1/3sqrt(x).

3.9-45 §3.9 source ↗

For the following exercises, use a calculator to graph the equation implied by the given variation.

$y$ varies inversely as the square of $x$ and when $x = 1, y = 4.$

Answer

$y = \frac{4}{x^{2}}$

A graph of a function with a vertical dashed asymptote at x=0 and a horizontal dashed asymptote at y=0. The blue curve is symmetric about the y-axis, approaching infinity near x=0 and zero as x extends horizontally.

3.9-47 §3.9 source ↗

For the following exercises, use Kepler’s Law, which states that the square of the time, $T,$ required for a planet to orbit the Sun varies directly with the cube of the mean distance, $a,$ that the planet is from the Sun.

Use the result from the previous exercise to determine the time required for Mars to orbit the Sun if its mean distance is 142 million miles.

Answer

≈ 1.89 years

3.9-49 §3.9 source ↗

For the following exercises, use Kepler’s Law, which states that the square of the time, $T,$ required for a planet to orbit the Sun varies directly with the cube of the mean distance, $a,$ that the planet is from the Sun.

Use the result from the previous exercise to determine the time required for Venus to orbit the Sun if its mean distance is 108 million kilometers.

Answer

≈ 0.61 years

3.9-51 §3.9 source ↗

For the following exercises, use the given information to answer the questions.

The distance $s$ that an object falls varies directly with the square of the time, $t,$ of the fall. If an object falls 16 feet in one second, how long for it to fall 144 feet?

Answer

3 seconds

3.9-53 §3.9 source ↗

For the following exercises, use the given information to answer the questions.

The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?

Answer

48 inches

3.9-55 §3.9 source ↗

For the following exercises, use the given information to answer the questions.

The weight of an object above the surface of the Earth varies inversely with the square of the distance from the center of the Earth. If a body weighs 50 pounds when it is 3960 miles from Earth’s center, what would it weigh it were 3970 miles from Earth’s center?

Answer

≈ 49.75 pounds

3.9-57 §3.9 source ↗

For the following exercises, use the given information to answer the questions.

The current in a circuit varies inversely with its resistance measured in ohms. When the current in a circuit is 40 amperes, the resistance is 10 ohms. Find the current if the resistance is 12 ohms.

Answer

≈ 33.33 amperes

3.9-59 §3.9 source ↗

For the following exercises, use the given information to answer the questions.

The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm)) and the cube of the diameter. If the shaft of a certain material 3 inches in diameter can transmit 45 hp at 100 rpm, what must the diameter be in order to transmit 60 hp at 150 rpm?

Answer

≈ 2.88 inches

3.R-1 §3.R source ↗

You have reached the end of Chapter 3: Polynomial and Rational Functions. Let’s review some of the Key Terms, Concepts and Equations you have learned.

Perform the indicated operation with complex numbers.

$(4 + 3 i) +$ $(- 2 - 5 i)$

Answer

2 - 2 i

3.R-3 §3.R source ↗

You have reached the end of Chapter 3: Polynomial and Rational Functions. Let’s review some of the Key Terms, Concepts and Equations you have learned.

Perform the indicated operation with complex numbers.

$(2 - 3 i) (3 + 6 i)$

Answer

24 + 3 i

3.R-5 §3.R source ↗

Solve the following equations over the complex number system.

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

Answer

{2 + i, 2 - i}

3.R-7 §3.R source ↗

For the following exercises, write the quadratic function in standard form. Then, give the vertex and axes intercepts. Finally, graph the function.

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

Answer

$f (x) =$ ${(x - 2)}^{2} -$ $9 vertex (2, –9),$ $intercepts (5, 0);$ $(–1, 0);$ $(0, –5)$

Graph of f(x)=x^2-4x-5.

3.R-9 §3.R source ↗

For the following problems, find the equation of the quadratic function using the given information.

The vertex is $(- 2, 3)$ and a point on the graph is $(3, 6) .$

Answer

f (x) =

\frac{3}{25} {(x + 2)}^{2} +

3

3.R-11 §3.R source ↗

Answer the following questions.

A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the dimensions of the plot to have maximum area.

Answer

300 meters by 150 meters, the longer side parallel to river.

3.R-13 §3.R source ↗

For the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.

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

Answer

Yes, degree = 5, leading coefficient = 4

3.R-15 §3.R source ↗

For the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.

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

Answer

Yes, degree = 4, leading coefficient = 1

3.R-17 §3.R source ↗

For the following exercises, determine end behavior of the polynomial function.

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

Answer

As x \to

-

\infty,

f (x) \to

-

\infty,

as x \to

\infty,

f (x) \to

\infty

3.R-19 §3.R source ↗

For the following exercises, find all zeros of the polynomial function, noting multiplicities.

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

Answer

–3 with multiplicity 2,

\frac{1}{2}

with multiplicity 1, –1 with multiplicity 3

3.R-21 §3.R source ↗

For the following exercises, find all zeros of the polynomial function, noting multiplicities.

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

Answer

4 with multiplicity 1

3.R-23 §3.R source ↗

For the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.

Graph of an even-degree polynomial with two turning points.

Answer

\frac{1}{2}

with multiplicity 1, 3 with multiplicity 3

3.R-25 §3.R source ↗

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

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

Answer

x^{2} + 4

with remainder 12

3.R-27 §3.R source ↗

For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

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

Answer

x^{2} -

5 x +

20 -

\frac{61}{x + 3}

3.R-29 §3.R source ↗

For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

$\frac{2 x^{3} + 6 x^{2} - 11 x - 12}{x + 4}$

Answer

2 x^{2} - 2 x - 3

, so factored form is

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

3.R-31 §3.R source ↗

For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.

$2 x^{3} -$ $3 x^{2} -$ $18 x -$ $8 =$ $0$

Answer

{- 2, 4, - \frac{1}{2}}

3.R-33 §3.R source ↗

For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.

$2 x^{4} -$ $17 x^{3} +$ $46 x^{2} -$ $43 x +$ $12 =$ $0$

Answer

{1, 3, 4, \frac{1}{2}}

3.R-35 §3.R source ↗

For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.

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

Answer

0 or 2 positive, 1 negative

3.R-37 §3.R source ↗

For the following rational functions, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph.

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

Answer

Intercepts $(–2, 0) and (0, - \frac{2}{5})$ , Asymptotes $x = 5$ and $y = 1.$

Graph of f(x)=(x+1)/(x-5).

3.R-39 §3.R source ↗

For the following rational functions, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph.

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

Answer

Intercepts (3, 0), (-3, 0), and $(0, \frac{27}{2})$ , Asymptotes $x =$ $1,$ $x =$ $- 2,$ $y =$ $3.$

Graph of f(x)=(3x^2-27)/(x^2+x-2).

3.R-41 §3.R source ↗

For the following exercises, find the slant asymptote.

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

Answer

y = x - 2

3.R-43 §3.R source ↗

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

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

Answer

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

3.R-45 §3.R source ↗

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

$f (x) =$ $x^{2} +$ $6 x -$ $2,$ $x \geq -$ $3$

Answer

f^{- 1} (x) =

\sqrt{x + 11} -

3

3.R-47 §3.R source ↗

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

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

Answer

f^{- 1} (x) =

\frac{{(x + 3)}^{2} - 5}{4},

x \geq -

3

3.R-49 §3.R source ↗

For the following exercises, find the unknown value.

$y$ varies directly as the square of $x .$ If when $x = 3, y = 36,$ find $y$ if $x = 4.$

Answer

y = 64

3.R-51 §3.R source ↗

For the following exercises, find the unknown value.

$y$ varies jointly as the cube of $x$ and as $z .$ If when $x = 1$ and $z = 2,$ $y = 6,$ find $y$ if $x = 2$ and $z = 3.$

Answer

y = 72

3.R-53 §3.R source ↗

For the following exercises, solve the application problem.

The weight of an object above the surface of the earth varies inversely with the square of the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.

Answer

148.5 pounds

3.T-1 §3.T source ↗

Perform the indicated operation or solve the equation.

$(3 - 4 i) (4 + 2 i)$

Answer

20 - 10 i

3.T-3 §3.T source ↗

Perform the indicated operation or solve the equation.

$x^{2} - 4 x + 13 = 0$

Answer

{2 + 3 i, 2 - 3 i}

3.T-5 §3.T source ↗

Determine the end behavior of the polynomial function.

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

Answer

A s x \to

-

\infty,

f (x) \to

-

\infty,

a s x \to

\infty,

f (x) \to

\infty

3.T-7 §3.T source ↗

Write the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.

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

Answer

$f (x) =$ ${(x + 1)}^{2} -$ $9$ , vertex $(−1, −9)$ , intercepts $(2, 0);$ $(−4, 0);$ $(0, −8)$

Graph of f(x)=x^2+2x-8.

3.T-9 §3.T source ↗

Solve the following application problem.

A rectangular field is to be enclosed by fencing. In addition to the enclosing fence, another fence is to divide the field into two parts, running parallel to two sides. If 1,200 feet of fencing is available, find the maximum area that can be enclosed.

Answer

60,000 square feet

3.T-11 §3.T source ↗

Find all zeros of the following polynomial functions, noting multiplicities.

$f (x) =$ $2 x^{6} -$ $12 x^{5} +$ $18 x^{4}$

Answer

0 with multiplicity 4, 3 with multiplicity 2

3.T-13 §3.T source ↗

Use long division to find the quotient.

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

Answer

2 x^{2} -

4 x +

11 -

\frac{26}{x + 2}

3.T-15 §3.T source ↗

Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.

$\frac{2 x^{3} + 5 x^{2} - 7 x - 12}{x + 3}$

Answer

2 x^{2} - x - 4

. So factored form is

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

3.T-17 §3.T source ↗

Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.

$f (x) =$ $4 x^{4} +$ $8 x^{3} +$ $21 x^{2} +$ $17 x +$ $4$

Answer

- \frac{1}{2}

(has multiplicity 2),

\frac{- 1 \pm i \sqrt{15}}{2}

3.T-19 §3.T source ↗

Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.

$f (x) =$ $x^{5} +$ $6 x^{4} +$ $13 x^{3} +$ $14 x^{2} +$ $12 x +$ $8$

Answer

- 2

(has multiplicity 3),

\pm i

3.T-21 §3.T source ↗

Given the following information about a polynomial function, find the function.

It has a zero of multiplicity 3 at $x = \frac{1}{2}$ and another zero at $x = - 3$ . It contains the point $(1, 8) .$

Answer

f (x) =

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

3.T-23 §3.T source ↗

For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.

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

Answer

Intercepts $(- 4, 0),$ $(0, - \frac{4}{3})$ , Asymptotes $x =$ $3,$ $x =$ $−1,$ $y =$ $0$ .

Graph of f(x)=(x+4)/(x^2-2x-3).

3.T-25 §3.T source ↗

Find the slant asymptote of the rational function.

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

Answer

y = x + 4

3.T-27 §3.T source ↗

Find the inverse of the function.

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

Answer

f^{- 1} (x) =

\sqrt[3]{\frac{x + 4}{3}}

3.T-29 §3.T source ↗

Find the unknown value.

$y$ varies inversely as the square of $x$ and when $x = 3,$ $y = 2.$ Find $y$ if $x = 1.$

Answer

y = 18

3.T-31 §3.T source ↗

Solve the following application problem.

The distance a body falls varies directly as the square of the time it falls. If an object falls 64 feet in 2 seconds, how long will it take to fall 256 feet?

Answer

4 seconds

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Problems and answers are adapted from Precalculus 2e by Jay Abramson, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/precalculus-2e/pages/1-introduction-to-functions. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.