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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

Why is b in the answer
Dahsolar Reply
how do you work it out?
Brad Reply
answer
Ernest
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Nitin
(Pcos∅+qsin∅)/(pcos∅-psin∅)
John Reply
how to do that?
Rosemary Reply
what is it about?
Amoah
how to answer the activity
Chabelita Reply
how to solve the activity
Chabelita
solve for X,,4^X-6(2^)-16=0
Alieu Reply
x4xminus 2
Lominate
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harish Reply
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
ZAHRO Reply
If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .
Swetha Reply
Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?
Patrick Reply
Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?
Patrick
how I can simplify algebraic expressions
Katleho Reply
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
Mary Reply
23yrs
Yeboah
lairenea's age is 23yrs
ACKA
hy
Katleho
Ello everyone
Katleho
Laurene is 46 yrs and Mae is 23 is
Solomon
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christopher
age does not matter
christopher
solve for X, 4^x-6(2*)-16=0
Alieu
prove`x^3-3x-2cosA=0 (-π<A<=π
Mayank Reply
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Rose Reply
Excusme but what are you wrot?
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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