Operations With Rational Expressions
A rational expression is an expression that can be written in the form
polynominal
polynominal
,
fraction polynominal , over polynominal end fraction . comma where the denominator is not zero. A rational expression is in simplest form if the numerator and denominator have no common factors except 1.
Example 1
Write the expression
4
x
+
8
x
+
2
fraction 4 x plus 8 , over x plus 2 end fraction in simplest form.
4
x
+
8
x
+
2
=
4
(
x
+
2
)
x
+
2
Factor the numerator
.
=
4
Divide out the common factor
x
+
2.
table with 2 rows and 3 columns , row1 column 1 , fraction 4 x plus 8 , over x plus 2 end fraction , column 2 equals . fraction 4 . open , x plus 2 , close , over x plus 2 end fraction , column 3 cap factorthenumerator . . , row2 column 1 , , column 2 equals 4 , column 3 cap divideoutthecommonfactor . x plus 2. , end table
To add or subtract two rational expressions, use a common denominator.
Example 2
Simplify
x
2
y
+
x
3
y
.
fraction x , over 2 y end fraction , plus , fraction x , over 3 y end fraction , .
x
2
y
+
x
3
y
=
x
2
y
⋅
3
3
+
x
3
y
⋅
2
2
The common denominator of
3
y
and
2
y
is
6
y
.
=
3
x
6
y
+
2
x
6
y
=
5
x
6
y
Add the numerators
.
table with 3 rows and 3 columns , row1 column 1 , fraction x , over 2 y end fraction , plus , fraction x , over 3 y end fraction , column 2 equals , fraction x , over 2 y end fraction , dot , 3 thirds , plus , fraction x , over 3 y end fraction , dot , 2 halves , column 3 cap thecommondenominatorof . 3 y , and , 2 y , is , 6 y . , row2 column 1 , , column 2 equals . fraction 3 x , over 6 y end fraction . plus . fraction 2 x , over 6 y end fraction , column 3 , row3 column 1 , , column 2 equals . fraction 5 x , over 6 y end fraction , column 3 cap addthenumerators . . , end table
To multiply rational expressions, first find and divide out any common factors in the numerators and the denominators. Then multiply the remaining numerators and denominators. To divide rational expressions, first use a reciprocal to change the problem to multiplication.
Example 3
Simplify
40
x
2
21
÷
5
x
14
.
fraction 40 , x squared , over 21 end fraction . divides , fraction 5 x , over 14 end fraction , .
40
x
2
21
÷
5
x
14
=
40
x
2
21
⋅
14
5
x
Change dividing by
5
x
14
to multiplying by the reciprocal
,
14
5
x
.
=
8
40
x
2
1
3
21
×
14
2
5
x
1
Divide out the common factors
5
,
x
,
and
7.
=
16
x
3
Multiply the numerators
(
8
x
⋅
2
)
.
Multiply the denominators
(
3
⋅
1
)
.
table with 3 rows and 3 columns , row1 column 1 , fraction 40 , x squared , over 21 end fraction . divides , fraction 5 x , over 14 end fraction , column 2 equals . fraction 40 , x squared , over 21 end fraction . dot , fraction 14 , over 5 x end fraction , column 3 cap changedividingby . fraction 5 x , over 14 end fraction . tomultiplyingbythereciprocal . comma , fraction 14 , over 5 x end fraction , . , row2 column 1 , , column 2 equals . fraction to the eighth , cross out enclosing , 40 , end crossout x , cross out enclosing , 2 , end crossout to the first , over sub 3 , cross out enclosing , 21 , end crossout end fraction . times . fraction cross out enclosing , 14 , end crossout squared , over
Exercises
Write each expression in simplest form.
-
4
a
2
b
12
a
b
3
fraction 4 , eh squared , b , over 12 eh , b cubed end fraction
-
5
n
+
15
n
+
3
fraction 5 n plus 15 , over n plus 3 end fraction
-
x
−
7
2
x
−
14
fraction x minus 7 , over 2 x minus 14 end fraction
-
28
c
2
(
d
−
3
)
35
c
(
d
−
3
)
fraction 28 , c squared . open , d minus 3 , close , over 35 c . open , d minus 3 , close end fraction
Perform the indicated operation.
-
3
x
2
+
5
x
2
fraction 3 x , over 2 end fraction , plus , fraction 5 x , over 2 end fraction
-
3
x
8
+
5
x
8
fraction 3 x , over 8 end fraction , plus , fraction 5 x , over 8 end fraction
-
5
h
−
3
h
5 over h , minus , 3 over h
-
6
11
p
−
9
11
p
fraction 6 , over 11 p end fraction , minus , fraction 9 , over 11 p end fraction
-
3
x
5
−
x
2
fraction 3 x , over 5 end fraction , minus , x over 2
-
13
2
x
−
13
3
x
fraction 13 , over 2 x end fraction , minus , fraction 13 , over 3 x end fraction
-
7
x
5
+
5
x
7
fraction 7 x , over 5 end fraction , plus , fraction 5 x , over 7 end fraction
-
5
a
b
+
3
a
5
b
fraction 5 eh , over b end fraction , plus . fraction 3 eh , over 5 b end fraction
-
7
x
8
⋅
32
x
35
fraction 7 x , over 8 end fraction , dot . fraction 32 x , over 35 end fraction
-
3
x
2
2
⋅
6
x
fraction 3 , x squared , over 2 end fraction . dot , 6 over x
-
8
x
2
5
⋅
10
x
3
fraction 8 , x squared , over 5 end fraction . dot , fraction 10 , over x cubed end fraction
-
7
x
8
⋅
64
14
x
fraction 7 x , over 8 end fraction , dot . fraction 64 , over 14 x end fraction
-
16
3
x
÷
5
3
x
fraction 16 , over 3 x end fraction , divides , fraction 5 , over 3 x end fraction
-
4
x
5
÷
16
15
x
fraction 4 x , over 5 end fraction , divides . fraction 16 , over 15 x end fraction
-
x
3
8
÷
x
2
16
fraction x cubed , over 8 end fraction , divides , fraction x squared , over 16 end fraction
