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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

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
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
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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