<< Chapter < Page Chapter >> Page >

f ( x ) = 2 x 3 9 x 2 + 13 x 6 ;   x 1

Got questions? Get instant answers now!

f ( x ) = 2 x 3 + x 2 5 x + 2 ;   x + 2

2 ,   1 ,   1 2

Got questions? Get instant answers now!

f ( x ) = 3 x 3 + x 2 20 x + 12 ;   x + 3

Got questions? Get instant answers now!

f ( x ) = 2 x 3 + 3 x 2 + x + 6 ; x + 2

2

Got questions? Get instant answers now!

f ( x ) = 5 x 3 + 16 x 2 9 ; x 3

Got questions? Get instant answers now!

x 3 + 3 x 2 + 4 x + 12 ; x + 3

3

Got questions? Get instant answers now!

4 x 3 7 x + 3 ; x 1

Got questions? Get instant answers now!

2 x 3 + 5 x 2 12 x 30 , 2 x + 5

5 2 ,   6 ,   6

Got questions? Get instant answers now!

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

x 3 3 x 2 10 x + 24 = 0

Got questions? Get instant answers now!

2 x 3 + 7 x 2 10 x 24 = 0

2 ,   4 ,   3 2

Got questions? Get instant answers now!

x 3 + 2 x 2 9 x 18 = 0

Got questions? Get instant answers now!

x 3 + 5 x 2 16 x 80 = 0

4 ,   4 ,   5

Got questions? Get instant answers now!

x 3 3 x 2 25 x + 75 = 0

Got questions? Get instant answers now!

2 x 3 3 x 2 32 x 15 = 0

5 ,   3 ,   1 2

Got questions? Get instant answers now!

2 x 3 + x 2 7 x 6 = 0

Got questions? Get instant answers now!

2 x 3 3 x 2 x + 1 = 0

1 2 ,   1 + 5 2 ,   1 5 2

Got questions? Get instant answers now!

3 x 3 x 2 11 x 6 = 0

Got questions? Get instant answers now!

2 x 3 5 x 2 + 9 x 9 = 0

3 2

Got questions? Get instant answers now!

2 x 3 3 x 2 + 4 x + 3 = 0

Got questions? Get instant answers now!

x 4 2 x 3 7 x 2 + 8 x + 12 = 0

2 ,   3 ,   1 ,   2

Got questions? Get instant answers now!

x 4 + 2 x 3 9 x 2 2 x + 8 = 0

Got questions? Get instant answers now!

4 x 4 + 4 x 3 25 x 2 x + 6 = 0

1 2 ,   1 2 ,   2 ,   3

Got questions? Get instant answers now!

2 x 4 3 x 3 15 x 2 + 32 x 12 = 0

Got questions? Get instant answers now!

x 4 + 2 x 3 4 x 2 10 x 5 = 0

1 ,   1 ,   5 ,   5

Got questions? Get instant answers now!

8 x 4 + 26 x 3 + 39 x 2 + 26 x + 6

3 4 ,   1 2

Got questions? Get instant answers now!

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

x 3 8 x 2 + 25 x 26 = 0

2 ,   3 + 2 i ,   3 2 i

Got questions? Get instant answers now!

x 3 + 13 x 2 + 57 x + 85 = 0

Got questions? Get instant answers now!

3 x 3 4 x 2 + 11 x + 10 = 0

2 3 ,   1 + 2 i ,   1 2 i

Got questions? Get instant answers now!

x 4 + 2 x 3 + 22 x 2 + 50 x 75 = 0

Got questions? Get instant answers now!

2 x 3 3 x 2 + 32 x + 17 = 0

1 2 ,   1 + 4 i ,   1 4 i

Got questions? Get instant answers now!

Graphical

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

f ( x ) = x 4 x 2 1

1 positive, 1 negative

Graph of f(x)=x^4-x^2-1.
Got questions? Get instant answers now!

f ( x ) = x 3 2 x 2 5 x + 6

Got questions? Get instant answers now!

f ( x ) = x 3 2 x 2 + x 1

3 or 1 positive, 0 negative

Graph of f(x)=x^3-2x^2+x-1.
Got questions? Get instant answers now!

f ( x ) = x 4 + 2 x 3 12 x 2 + 14 x 5

Got questions? Get instant answers now!

f ( x ) = 2 x 3 + 37 x 2 + 200 x + 300

0 positive, 3 or 1 negative

Graph of f(x)=2x^3+37x^2+200x+300.
Got questions? Get instant answers now!

f ( x ) = x 3 2 x 2 16 x + 32

Got questions? Get instant answers now!

f ( x ) = 2 x 4 5 x 3 5 x 2 + 5 x + 3

2 or 0 positive, 2 or 0 negative

Graph of f(x)=2x^4-5x^3-5x^2+5x+3.
Got questions? Get instant answers now!

f ( x ) = 2 x 4 5 x 3 14 x 2 + 20 x + 8

Got questions? Get instant answers now!

f ( x ) = 10 x 4 21 x 2 + 11

2 or 0 positive, 2 or 0 negative

Graph of f(x)=10x^4-21x^2+11.
Got questions? Get instant answers now!

Numeric

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

f ( x ) = x 4 + 3 x 3 4 x + 4

Got questions? Get instant answers now!

f ( x ) = 2 x 3 + 3 x 2 8 x + 5

± 5 ,   ± 1 ,   ± 5 2

Got questions? Get instant answers now!

f ( x ) = 3 x 3 + 5 x 2 5 x + 4

Got questions? Get instant answers now!

f ( x ) = 6 x 4 10 x 2 + 13 x + 1

± 1 ,   ± 1 2 ,   ± 1 3 ,   ± 1 6

Got questions? Get instant answers now!

f ( x ) = 4 x 5 10 x 4 + 8 x 3 + x 2 8

Got questions? Get instant answers now!

Technology

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

1 ,   1 2 ,   1 3

Got questions? Get instant answers now!

f ( x ) = 4 x 3 4 x 2 13 x 5

Got questions? Get instant answers now!

f ( x ) = 8 x 3 6 x 2 23 x + 6

2 ,   1 4 ,   3 2

Got questions? Get instant answers now!

f ( x ) = 12 x 4 + 55 x 3 + 12 x 2 117 x + 54

Got questions? Get instant answers now!

f ( x ) = 16 x 4 24 x 3 + x 2 15 x + 25

5 4

Got questions? Get instant answers now!

Extensions

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

Real roots: –1, 1, 3 and ( 2 , f ( 2 ) ) = ( 2 , 4 )

Got questions? Get instant answers now!

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

f ( x ) = 4 9 ( x 3 + x 2 x 1 )

Got questions? Get instant answers now!

Real roots: –2, 1 2 (with multiplicity 2) and ( 3 , f ( 3 ) ) = ( 3 , 5 )

Got questions? Get instant answers now!

Real roots: 1 2 , 0, 1 2 and ( 2 , f ( 2 ) ) = ( 2 , 6 )

f ( x ) = 1 5 ( 4 x 3 x )

Got questions? Get instant answers now!

Real roots: –4, –1, 1, 4 and ( 2 , f ( 2 ) ) = ( 2 , 10 )

Got questions? Get instant answers now!

Real-world applications

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.

8 by 4 by 6 inches

Got questions? Get instant answers now!

The length, width, and height are consecutive whole numbers. The volume is 120 cubic inches.

Got questions? Get instant answers now!

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

5.5 by 4.5 by 3.5 inches

Got questions? Get instant answers now!

The length is three times the height and the height is one inch less than the width. The volume is 108 cubic inches.

Got questions? Get instant answers now!

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

8 by 5 by 3 inches

Got questions? Get instant answers now!

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

The radius is 3 inches more than the height. The volume is 16 π cubic meters.

Got questions? Get instant answers now!

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

Radius = 6 meters, Height = 2 meters

Got questions? Get instant answers now!

The radius and height differ by one meter. The radius is larger and the volume is 48 π cubic meters.

Got questions? Get instant answers now!

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

Radius = 2.5 meters, Height = 4.5 meters

Got questions? Get instant answers now!

80. The radius is 1 3 meter greater than the height. The volume is 98 9 π cubic meters.

Got questions? Get instant answers now!

Questions & Answers

the diagram of the digestive system
Assiatu Reply
allimentary cannel
Ogenrwot
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 6

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask