Adding and subtracting rational expressions:

Adding and subtracting rational expressions:

To add or subtract rational expressions use the addition property:

b

ca

b

c

b

a

Taken from http://podcasts.shelbyed.k12.al.us/sclemons/files/2011/05/add-subtraction-rationals.ppt#1

If the expressions have the same (common) denominator, add or

subtract the numerators, keep the same denominator, and simplify

the result.

x

a

x

a

x

a

x

a 2

3

6

3

4

3

2


Do the operations and simplify the result:

4

5

4

xx

x

4

5

x

x


4

3

4

2

4

4

x

x

x

x

x

4

4

x

x

=1)4(

)4(

x

x


1213

3

1213

4222

xx

x

xx

x

1213

)3(422

xx

xx

1213

12

xx

x)12)(1(

)1(

xx

x

12

1

x


yx

y

yx

x

99

yx

yx

99

yx

yx

)(9

=9Taken from http://podcasts.shelbyed.k12.al.us/sclemons/files/2011/05/add-subtraction-rationals.ppt#1

xy

xy

yx

2323

xy

y

xyx

2323

)2)(2(

5)2(4)2(3

)2)(2(

5

)2(

4

)2(

3

xx

xx

xxxx


To add or subtract fractions with different denominators find the least common denominator (LCD), change

each fraction so that it has that denominator then add or subtract

Simply stated, the LCD is the least number (expression) that all denominators

will divide into evenly.


To find the LCD

• factor each denominator• write the different factors• give each factor the highest power

to which it occurs• multiply the results


Find the LCD for y²-16 and y²-8y+16

• Factor each• write each factor• give each highest exponent• multiply results

• (y+4)(y-4) and (y-4)²

• (y+4)(y-4)

• (y+4)(y-4)²• (y+4)(y-4)²


Add : m

n

n

m

4

3

2

5

• LCD =• change each

fraction to LCD

• add/subtract• simplify

mn4

mn

nn

mn

mm

4

)(3

4

)2(5

mn

nm

4

310 22


Add :


fraction to LCD


1

12

x

1x

1

1

1

)1(2

xx

x

1

122

x

x

1

12

x

x


Add :


fraction to LCD


6

4

3

7

x

x

x

63 xx

63

34

63

67

xx

xx

xx

x

)6)(3(

42194

63

124427 22

xx

xx

xx

xxx

(Top isn’t factorable)


If the denominators have opposite factors, then change one

of them using the 3 signs of a fraction rule.

b

a

b

a

b

a


2

42

x

x

2

1

2

3

x

x

x

x

2

1

)2(

)3(

x

x

x

x

2

1

2

3

x

x

x

x

2)2(

)2(2

x

x


1)

24

8

2

2

2 yyy

y

2)

107

32

152

53

6

12222

xx

x

xx

x

xx

x

3)

9

32

x

x

4)

121

2

1

222

xx

x

x

x

x


9

32

x

x

)9(

3)9()9(2

x

xxx

)9(

39182 2

x

xxx

)9(

15112

x

xx


107

32

152

53

6

12222

xx

x

xx

x

xx

x

(x+3) (x-2) (x+3) (x-5) (x-5) (x-2)

)5)(2)(3(

)3)(32())2)(53(()5)(12(

xxx

xxxxxx

)5)(2)(3(

)932()10113()5112( 222

xxx

xxxxxx

)5)(2)(3(

)143( 2

xxx

xx


24

8

2

2

2 yyy

y

(y – 2) (y+2) -(y-2)(y+2)

)2)(2(

)8)(1()2)(2()2(

yy

yyy

)2)(2(

84222

yy

yyy

)2)(2(

42

yy

y1

)2)(2(

)2)(2(

yy

yy

121

2

1

222

xx

x

x

x

x(x-1) (x-1)(x+1) (x+1)(x+1)

LCM: (x-1)(x+1)(x+1)

)1)(1)(1(

)1()1(2)1)(1(2

xxx

xxxxxx

)1)(1)(1(

22)12(2 222

xxx

xxxxxx

)1)(1)(1(

)23(1

)1)(1)(1(

23 22

xxx

xx

xxx

xx


xy

yx

3

3

Simplify the top and bottom:

yxyxxy

3

3

x

y

xxy

yxy

)3(

)3(Use “tortilla method”

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Adding and subtracting rational expressions: