Algebraic Fractions.docx

7/26/2019 Algebraic Fractions.docx

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COLLEGE ALGEBRA

Topic Rational Algebraic Fractions

Introduction

In arithmetic, the concepts and skills of fractions have been restricted to the ratio of

counting number. Fractions involving variables also arise in algebra. They are called rational

algebraic expression or simply algebraic fractions to distinguish them from arithmetic fractions.

Since division by ero is a restriction, the variables in rational algebraic expressions may nottake values !hich make the denominator e"ual to ero.

Revie! Fractions

A rational number of the forma

b !here b0 , is called a fraction. The "uantities a

and b are the terms of the fraction. The term a is called the numerator and the term b is called

the denominator.

a. Simplifying fractions

A fraction is said to be reduced in lo!est terms !hen the greatest common factor #$%F&of its numerator and denominator is '.To reduce fractions to lo!est terms,'& Find the $%F of the numerator and denominator.(& )ivide both the numerator and denominator by the $%F.

*xample+ Reduce20

24 to its lo!est term. Solution+ the $%F of the ( and

(- is -.

20

24=

244

244=

5

6

b. ultiplying fractionsIn multiplying t!o or more fractions, simply multiply their numerators, and then multiply

their denominators. Simplify the product if possible. /ote0 express mixed fractions as

improper fraction1efore performing the operation.

In symbols, if

a

bc

dare fractions, then

a

bc

d =

ac

bd

c. )ividing FractionsIn dividing a fraction by another, multiply the reciprocal of the divisor by the

dividend. Simplify the "uotient if possible. /ote+ *xpress mixed fractions as

improper fractions before performing the operation.

In symbols, if

a

bc

dare fraction, then a

bc

d=

a

bd

c =

ad

bc

d. Adding and subtracting Fractions


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/ote+ express mixed fractions as improper fractions before performing the

operation. Adding and Subtracting similar fractions+

T!o or more fractions are similar if they have the same denominator. 2nly similar

fractions can be added or subtracted.to combine similar fractions, combine the numerators and keep the common

denominator. In symbols, if

a

b

c

b are fractions, then

a

b+

c

b=

a+c

b , and

a

b

c

b=

acb .

Adding and subtracting dissimilar fractions+To add or subtract dissimilar fractions, first, re!rite them as similar fractions. In

symbols, if

a

bc

dare fractions and e is the least common denominator #3%)&,

then a

bc

d=

ae

b ce

d

e.

The 3%) is the least common multiple #3%& of the denominators of the fraction.

3% is the smallest natural number for !hich the both denominators are factors.

*xample+5

8+ 5

12=15+1024

=25

24

e. Simplifying complex Fractions

%omplex fractions have fractions in their numerator or denominator, or in both. A complex fraction can be simplified by individually simplifying the terms of the

complex fractons to obtain a single fraction.

*xample simplify

2

3+4

6

53

4

solution+

2

3+4

6

53

4

=

4+46

2034

=

8

6

17

4

=

8

6

3

24

17=

16

51

A complex fraction can also be simplified by multiplying both terms of the complexfraction by the 3%) of all the fractions in the complex fraction.

*xample+ simplify

2

3+ 46

53

4

Solution+

2

3+ 46

53

4

=( 23+ 46)12

(43

4)12

= 8+8609

=16

51

*xercises+a& 4erform the indicated operations. *xpress the final ans!ers as fractions in lo!est terms.

'.4

12 7

12 (.5

4 7

16 5. 4

2

32

1

5


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-.12

17+

5

17

6. 115

6+9

7

5

7.3

71

5

3

8.

7

84

5(467 )

9.3

512

35

:. 25

87

1

2

'.4

7(36)

''.

(2 38 )(3 512 ) 512

'(. 2

2

7

+35

9

b& Simplify each complex fraction.

'&

1

2+3

5

3

41

5

(&

5

10

5

3

1 36+ 23

5&

34

3

4+3

4

-&

4

5

3

8

5

67

8

6&

11

124

3313

4 39

64

7&

2

1+1

2

3

11

3

8& 1+1 12

1+2

3

9&

5

63

8

5

63

8


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:& Simplifying rational algebraic *xpressions

'& Algebraic expressions of the formn

d !here nand dare polynomials, !here

d 0 , is called a rational expressions. It is emphasied that the denominator of a rational

expression cannot be e"ual to the value of the variable that !ill make the denominator ero

is excluded.

''& A rational expression is said to be in lo!est terms !hen its numerator and

denominator have no common factors other than '. In reducing a rational expression to

lo!est terms, divide both terms of the rational expression by the $%F of the terms.

'(&*xamples+ Reduce the follo!ing rational expressions to lo!est terms.

'.10x

3

y5

z

12x3

y7

z2 Solution+ $%F; 2x

3

y5

z

'5&10x

4

y5

z2x3

y5

z

12x3

y7

z2

2x3

y5

z=

5x

6y2

z

(.10x

35x2

15x7 Solution+ Factor both terms of the rational expression, then divide out the

common factor.

'-&10x

35x2

15x7 =

5x2 (2x1 )

5x2 (3x )5 =

2x13x

5.x

236x6 Solution+ Factor both terms of the rational expression, then divide out the

common factor.

'6&x

236x6

=(x+6)(2x1)

x6=x+6

-.3y

2+7y6y

2+y6 Solution+ Factor both terms of the rational expression, then divide out the

common factor.

'7&3y

2+7y6y

2+y6=(3y2)(y+3)(y2)(x+5)

=3y2y2

6.y

2y69y2 solution+

y2y69y2 =

(y+2)(y3)(37)(3+y )=

(y+2)(y3)(1)(y3)(3+y)=

y+23+y

17)

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