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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 sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit 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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