Digital Lesson
Operations on Rational
Expressions
Rational expressions are fractions in which the
numerator and denominator are polynomials and the
denominator does not equal zero.
x 9
2
Example: Simplify


x 3
.
( x  3)( x  3)
x 3
( x  3)( x  3) ,
( x  3)
 ( x  3) ,
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x–30
x3
2
To multiply rational expressions:
1. Factor the numerator and denominator of each
fraction.
2. Multiply the numerators and denominators of
each fraction.
3. Divide by the common factors.
4. Write the answer in simplest form.
a c
•
b d
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
ac
bd
3
x  3x
2
Example: Multiply




x( x  3)
( x  3)( x  1)
•
x  2x  3
2
x x2
2
•
x  2x  3
2
.
( x  1)( x  2) Factor the numerator and
( x  3)( x  1) denominator of each fraction.
x( x  3)( x  1)( x  2)
( x  3)( x  1)( x  3)( x  1)
x( x  3)( x  1)( x  2)
( x  3)( x  1)( x  3)( x  1)
x( x  2)
( x  1)( x  3)
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Multiply.
Divide by the common factors.
Write the answer in simplest
form.
4
To divide rational expressions:
1. Multiply the dividend by the reciprocal of the
divisor. The reciprocal of
a
is
b
b
.
a
2. Multiply the numerators. Then multiply the
denominators.
3. Divide by the common factors.
4. Write the answer in simplest form.
a
b

c
d
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
a d
•
b c

ad
bc
5
xx y
2
Example: Divide
2x  2x y
2

z
xx y
z
2

z



z
•
2
Factor and multiply.
2
2 x(1  xy ) • z
z
Multiply by the reciprocal
of the divisor.
2
2 x(1  xy ) • z
x(1  xy ) • z
2
2
2x  2x y
x(1  xy ) • z
.
Divide by the common
factors.
Simplest form
2
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6
The least common multiple (LCM) of two or more numbers is the
least number that contains the prime factorization of each number.
Examples: 1. Find the LCM of 10 and 4.
10 = (5 • 2)
factors of 10
4 = (2 • 2)
LCM = 2 • 2 • 5 = 20
factors of 4
2. Find the LCM of 4x2 + 4x and x2 + 2x + 1.
4x2 + 4x = (4x)(x +1) = 2 • 2 x (x + 1)
x2 + 2x + 1 = (x +1)(x +1)
factors of x2 + 2x + 1
LCM = 2 • 2 x (x +1)(x +1) = 4x3 + 8x2 + 4x
factors of 4x2 + 4x
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7
Fractions can be expressed in terms of the least common multiple
of their denominators.
Example: Write the fractions
x
4x
2
and
2x 1
6 x  12 x
2
in terms of the
LCM of the denominators.
The LCM of the denominators is 12x2(x – 2).
x
4x
2
x

( 2 x)(2 x)
2x 1
6 x  12 x
2
•

3( x  2)
3( x  2)
2x 1
6 x( x  2)
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•
2x
2x

3( x  2)( x)
12 x ( x  2)

2
2 x( 2 x  1)
LCM
12 x ( x  2)
2
8
To add rational expressions:
1. If necessary, rewrite the fractions with a common
denominator.
2. Add the numerators of each fraction.
a

b
c

b
ac
b
To subtract rational expressions:
1. If necessary, rewrite the fractions with a common
denominator.
2. Subtract the numerators of each fraction.
a

b
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c
b

ac
b
9
Example: Add
2x
14


5x
.
14
2 x  5x

14
Example: Subtract
x 4
2
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
14
2x

7x

2x  4
x 4
2
x
2
4
x 4
2

.
2( x  2)
( x  2)( x  2)

2
( x  2)
10
Two rational expressions with different denominators can be
added or subtracted after they are rewritten with a common
denominator.
x 3
Example: Add
x  2x
2



6

x 3
x( x  2)
x 3
x( x  2)
.
x 4
2

•
6
( x  2)( x  2)
( x  2)
( x  2)
( x  3)( x  2)  6 x
x( x  2)( x  2)
x  5x  6
2

x( x  2)( x  2)
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

6
•
( x)
( x  2)( x  2) ( x)
x  x  6  6x
2

x( x  2)( x  2)
( x  6)( x  1)
x( x  2)( x  2)
11
Example: Subtract
x
2
x 1
2

1
x 1
2
x 1
.
2



Add numerators.
x 1
2
( x  1)( x  1)
( x  1)( x  1)
( x  1)( x  1)
( x  1)( x  1)
1
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Factor.
Divide.
Simplest form
12