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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

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het Reply
Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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