223 reference chapter section r5: rational expressions the quotient of two polynomials is called a...

223 Reference ChapterSection R5: Rational Expressions

The quotient of two polynomials is called a Rational Expression.

Determining the Domain: Remember, the denominator cannot be zero (that would make the fraction undefined.) So a rational expression would have a domain of all real numbers, with the exception of values that make the denominator zero.

Example: Find the domain of each of the following expressions1.

2.

3.

4

73

x

x

6

152 xx

7293

895 2

yy

yy


The quotient of two polynomials is called a Rational Expression.

Determining the Domain: Remember, the denominator cannot be zero (that would make the fraction undefined.) So a rational expression would have a domain of all real numbers, with the exception of values that make the denominator zero.

Example: Find the domain of each of the following expressions1. all real numbers except -4

2. All real numbers except -2, 3

3. All real numbers except -3, -7/2

4

73

x

x

6

152 xx

7293

895 2

yy

yy


Simplifying Rational Expressions.Write each numerator and denominator in terms of its factors, then simplify.

Example: Write each expression in lowest terms.1.

2.

3.

2010

42

x

x

72426

1232

mm

m

6

22

2

rr

rr


Simplifying Rational Expressions.Write each numerator and denominator in terms of its factors, then simplify.

Example: Write each expression in lowest terms.1. 1/5

2. 1 2m + 6

3. r + 1 r + 3

2010

42

x

x

72426

1232

mm

m

6

22

2

rr

rr


Simplifying Rational Expressions involving Multiplication.Write each numerator and denominator in terms of its factors, cancel out common factors, and simplify.

Example: Find each product.1.

2.

3.

6

3010

5

822

22

aa

aa

a

aa

6

86

12

92

2

2

2

cc

cc

cc

c

xx

xx

xx

xx

244

3011

20

42

2

2

23


Simplifying Rational Expressions involving Multiplication.Write each numerator and denominator in terms of its factors, cancel out common factors, and simplify.

Example: Find each product.1. 2a + 8

2. c + 4 c - 4

3. x 4

6

3010

5

822

22

aa

aa

a

aa

6

86

12

92

2

2

2

cc

cc

cc

c

xx

xx

xx

xx

244

3011

20

42

2

2

23


Simplifying Rational Expressions involving Division.First, change division to multiplication by finding the reciprocal of the second rational expression; then, write each numerator and denominator in terms of its factors, cancel out common factors, and simplify.Example: Find each product.1.

2.

3.

bb

bb

bb

b

182

3

98

92

2

2

2

34

32

1

122

2

2

2

zz

zz

z

zz

127

123

9

2722

3

dd

d

d

d


Simplifying Rational Expressions involving Division.First, change division to multiplication by finding the reciprocal of the second rational expression; then, write each numerator and denominator in terms of its factors, cancel out common factors, and simplify.Example: Find each product.1. 2b - 6 b + 1

2. 1

3. d^2 + 3d + 9 3

bb

bb

bb

b

182

3

98

92

2

2

2

34

32

1

122

2

2

2

zz

zz

z

zz

127

123

9

2722

3

dd

d

d

d


Simplifying Rational Expressions involving Addition and/or Subtraction.First, factor all of the numerators and denominators and determine the Least Common Denominator (LCD). Use the LCD to make both fractions have the same denominator, then add and/or subtract the fractions and simplify.Example: Find each sum or difference.

1.

2.

416

22

x

x

x

x

y

y

y

y

6

2

3

122

2


Simplifying Rational Expressions involving Addition and/or Subtraction.First, factor all of the numerators and denominators and determine the Least Common Denominator (LCD). Use the LCD to make both fractions have the same denominator, then add and/or subtract the fractions and simplify.Example: Find each sum or difference.

1. x^2 – 2x x^2 - 16

2. 3y^2 – 2y - 2 6y^2

416

22

x

x

x

x

y

y

y

y

6

2

3

122

2


Simplifying Rational Expressions involving Addition and/or Subtraction.Example: Find each sum or difference.

3.

4.

25

3

56

422

mmm

aaa

a

a

a 5

1

422


Simplifying Rational Expressions involving Addition and/or Subtraction.Example: Find each sum or difference.

3. 7m + 17 m^3 – m^2 – 25 + 1

4. -3a^2 + 3a + 5 a^3 - a

25

3

56

422

mmm

aaa

a

a

a 5

1

422


Simplifying Complex Fractions.The quotient of two rational expressions is referred to as a Complex Fraction. To simplify, multiply both the numerator and denominator by the LCD of all of the fractions.Example: Simplify the expression.

2

2

41

85

cc

cc


Simplifying Complex Fractions.The quotient of two rational expressions is referred to as a Complex Fraction. To simplify, multiply both the numerator and denominator by the LCD of all of the fractions.Example: Simplify the expression.

5c + 8 c - 4

2

2

41

85

cc

cc


Simplifying Complex Fractions.Example: Simplify the expression.

k

k2

1

56


Simplifying Complex Fractions.Example: Simplify the expression.

6k - 5 k + 2

k

k2

1

56

223 reference chapter section r5: rational expressions the quotient of two polynomials is called a...

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