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For the following exercises, divide the rational expressions.

3 y 2 7 y 6 2 y 2 3 y 9 ÷ y 2 + y 2 2 y 2 + y 3

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6 p 2 + p 12 8 p 2 + 18 p + 9 ÷ 6 p 2 11 p + 4 2 p 2 + 11 p 6

p + 6 4 p + 3

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q 2 9 q 2 + 6 q + 9 ÷ q 2 2 q 3 q 2 + 2 q 3

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18 d 2 + 77 d 18 27 d 2 15 d + 2 ÷ 3 d 2 + 29 d 44 9 d 2 15 d + 4

2 d + 9 d + 11

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16 x 2 + 18 x 55 32 x 2 36 x 11 ÷ 2 x 2 + 17 x + 30 4 x 2 + 25 x + 6

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144 b 2 25 72 b 2 6 b 10 ÷ 18 b 2 21 b + 5 36 b 2 18 b 10

12 b + 5 3 b −1

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16 a 2 24 a + 9 4 a 2 + 17 a 15 ÷ 16 a 2 9 4 a 2 + 11 a + 6

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22 y 2 + 59 y + 10 12 y 2 + 28 y 5 ÷ 11 y 2 + 46 y + 8 24 y 2 10 y + 1

4 y −1 y + 4

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9 x 2 + 3 x 20 3 x 2 7 x + 4 ÷ 6 x 2 + 4 x 10 x 2 2 x + 1

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For the following exercises, add and subtract the rational expressions, and then simplify.

4 x + 10 y

10 x + 4 y x y

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4 a + 1 + 5 a 3

9 a 7 a 2 2 a 3

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y + 3 y 2 + y 3 y + 1

2 y 2 y + 9 y 2 y 2

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x 1 x + 1 2 x + 3 2 x + 1

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3 z z + 1 + 2 z + 5 z 2

5 z 2 + z + 5 z 2 z 2

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x x + 1 + y y + 1

x + 2 x y + y x + x y + y + 1

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For the following exercises, simplify the rational expression.

2 a + 7 b b

2 b + 7 a a b 2

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3 a + b 6 2 b 3 a

18 + a b 4 b

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3 x + 1 + 2 x 1 x 1 x + 1

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a b b a a + b a b

a b

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2 c c + 2 + c 1 c + 1 2 c + 1 c + 1

3 c 2 + 3 c 2 2 c 2 + 5 c + 2

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Real-world applications

Brenda is placing tile on her bathroom floor. The area of the floor is 15 x 2 8 x 7 ft 2 . The area of one tile is x 2 2 x + 1 ft 2 . To find the number of tiles needed, simplify the rational expression: 15 x 2 8 x 7 x 2 2 x + 1 .

A rectangle that’s labeled: Area = fifteen times x squared minus eight times x minus seven.

15 x + 7 x −1

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The area of Sandy’s yard is 25 x 2 625 ft 2 . A patch of sod has an area of x 2 10 x + 25 ft 2 . Divide the two areas and simplify to find how many pieces of sod Sandy needs to cover her yard.

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Aaron wants to mulch his garden. His garden is x 2 + 18 x + 81 ft 2 . One bag of mulch covers x 2 81 ft 2 . Divide the expressions and simplify to find how many bags of mulch Aaron needs to mulch his garden.

x + 9 x −9

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Extensions

For the following exercises, perform the given operations and simplify.

x 2 + x 6 x 2 2 x 3 2 x 2 3 x 9 x 2 x 2 ÷ 10 x 2 + 27 x + 18 x 2 + 2 x + 1

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3 y 2 10 y + 3 3 y 2 + 5 y 2 2 y 2 3 y 20 2 y 2 y 15 y 4

1 y + 2

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4 a + 1 2 a 3 + 2 a 3 2 a + 3 4 a 2 + 9 a

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x 2 + 7 x + 12 x 2 + x 6 ÷ 3 x 2 + 19 x + 28 8 x 2 4 x 24 ÷ 2 x 2 + x 3 3 x 2 + 4 x 7

4

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Chapter review exercises

Real Numbers: Algebra Essentials

For the following exercises, perform the given operations.

( 5 3 2 ) 2 6

−5

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64 ÷ ( 2 8 ) + 14 ÷ 7

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For the following exercises, solve the equation.

For the following exercises, simplify the expression.

9 ( y + 2 ) ÷ 3 2 + 1

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3 m ( 4 + 7 ) m

32 m

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For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

Exponents and Scientific Notation

For the following exercises, simplify the expression.

6 a 2 a 0 2 a −4

a 6

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4 −2 x 3 y −3 2 x 0

x 3 32 y 3

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( 16 a 3 b 2 ) ( 4 a b −1 ) −2

a

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Write the number in standard notation: 2.1314 × 10 −6

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Write the number in scientific notation: 16,340,000

1.634 × 10 7

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Radicals and Rational Expressions

For the following exercises, find the principal square root.

Polynomials

For the following exercises, perform the given operations and simplify.

( 3 x 3 + 2 x 1 ) + ( 4 x 2 2 x + 7 )

3 x 3 + 4 x 2 + 6

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( 2 y + 1 ) ( 2 y 2 2 y 5 )

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( 2 x 2 + 3 x 6 ) + ( 3 x 2 4 x + 9 )

5 x 2 x + 3

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( 6 a 2 + 3 a + 10 ) ( 6 a 2 −3 a + 5 )

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( k + 3 ) ( k 6 )

k 2 3 k 18

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( 2 h + 1 ) ( 3 h 2 )

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( x + 1 ) ( x 2 + 1 )

x 3 + x 2 + x + 1

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( m 2 ) ( m 2 + 2 m 3 )

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( a + 2 b ) ( 3 a b )

3 a 2 + 5 a b 2 b 2

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Factoring Polynomials

For the following exercises, find the greatest common factor.

81 p + 9 p q 27 p 2 q 2

9 p

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12 x 2 y + 4 x y 2 −18 x y

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88 a 3 b + 4 a 2 b 144 a 2

4 a 2

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For the following exercises, factor the polynomial.

8 a 2 + 30 a 27

( 4 a 3 ) ( 2 a + 9 )

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x 2 + 10 x + 25

( x + 5 ) 2

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4 h 2 12 h k + 9 k 2

( 2 h 3 k ) 2

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p 3 + 216

( p + 6 ) ( p 2 6 p + 36 )

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64 q 3 27 p 3

( 4 q 3 p ) ( 16 q 2 + 12 p q + 9 p 2 )

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4 x ( x 1 ) 1 4 + 3 ( x 1 ) 3 4

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3 p ( p + 3 ) 1 3 −8 ( p + 3 ) 4 3

( p + 3 ) 1 3 ( −5 p 24 )

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4 r ( 2 r 1 ) 2 3 5 ( 2 r 1 ) 1 3

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Rational Expressions

For the following exercises, simplify the expression.

x 2 x 12 x 2 8 x + 16

x + 3 x 4

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4 y 2 25 4 y 2 20 y + 25

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2 a 2 a 3 2 a 2 6 a 8 5 a 2 19 a 4 10 a 2 13 a 3

1 2

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d 4 d 2 9 d 3 d 2 16

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m 2 + 5 m + 6 2 m 2 5 m 3 ÷ 2 m 2 + 3 m 9 4 m 2 4 m 3

m + 2 m 3

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4 d 2 7 d 2 6 d 2 17 d + 10 ÷ 8 d 2 + 6 d + 1 6 d 2 + 7 d 10

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10 x + 6 y

6 x + 10 y x y

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12 a 2 + 2 a + 1 3 a 2 −1

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1 d + 2 c 6 c + 12 d d c

1 6

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Chapter practice test

For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

For the following exercises, evaluate the equations.

2 ( x + 3 ) 12 = 18

x = 12

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y ( 3 + 3 ) 2 26 = 10

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Write the number in standard notation: 3.1415 × 10 6

3,141,500

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Write the number in scientific notation: 0.0000000212.

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For the following exercises, simplify the expression.

−2 ( 2 + 3 2 ) 2 + 144

16

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4 ( x + 3 ) ( 6 x + 2 )

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6 24 + 7 54 12 6

21 6

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( 13 q 3 + 2 q 2 3 ) ( 6 q 2 + 5 q 3 )

13 q 3 4 q 2 5 q

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( 6 p 2 + 2 p + 1 ) + ( 9 p 2 −1 )

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( n 2 ) ( n 2 4 n + 4 )

n 3 6 n 2 + 12 n 8

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( a 2 b ) ( 2 a + b )

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For the following exercises, factor the polynomial.

16 x 2 81

( 4 x + 9 ) ( 4 x 9 )

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27 c 3 1331

( 3 c 11 ) ( 9 c 2 + 33 c + 121 )

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3 x ( x 6 ) 1 4 + 2 ( x 6 ) 3 4

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For the following exercises, simplify the expression.

2 z 2 + 7 z + 3 z 2 9 4 z 2 15 z + 9 4 z 2 1

4 z 3 2 z 1

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a 2 b 2 b 9 a 3 a 2 b 6 a

3 a + 2 b 3 b

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Questions & Answers

the diagram of the digestive system
Assiatu Reply
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
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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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