Remember, the
reciprocal of
a
b is
b
a . To find the reciprocal we simply put the numerator in the denominator and the denominator in the numerator. We “flip” the fraction.
Division of rational expressions
If
p
,
q
,
r
,
s are polynomials where
q
≠
0
,
r
≠
0
,
s
≠
0 , then
p
q
÷
r
s
=
p
q
·
s
r
To divide rational expressions multiply the first fraction by the reciprocal of the second.
Divide rational expressions.
Rewrite the division as the product of the first rational expression and the reciprocal of the second.
Factor the numerators and denominators completely.
Multiply the numerators and denominators together.
Simplify by dividing out common factors.
Remember, first rewrite the division as multiplication of the first expression by the reciprocal of the second. Then factor everything and look for common factors.
Divide:
2
x
2
+
5
x
−
12
x
2
−
16
÷
2
x
2
−
13
x
+
15
x
2
−
8
x
+
16
.
Solution
2
x
2
+
5
x
−
12
x
2
−
16
÷
2
x
2
−
13
x
+
15
x
2
−
8
x
+
16
Rewrite the division as multiplication of
the first expression by the reciprocal of
the second.
2
x
2
+
5
x
−
12
x
2
−
16
·
x
2
−
8
x
+
16
2
x
2
−
13
x
+
15
Factor the numerators and denominators
and then multiply.
(
2
x
−
3
)
(
x
+
4
)
(
x
−
4
)
(
x
−
4
)
(
x
−
4
)
(
x
+
4
)
(
2
x
−
3
)
(
x
−
5
)
Simplify by dividing out common factors.
(
2
x
−
3
)
(
x
+
4
)
(
x
−
4
)
(
x
−
4
)
(
x
−
4
)
(
x
+
4
)
(
2
x
−
3
)
(
x
−
5
)
Simplify.
x
−
4
x
−
5
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Divide:
p
3
+
q
3
2
p
2
+
2
p
q
+
2
q
2
÷
p
2
−
q
2
6
.
Solution
p
3
+
q
3
2
p
2
+
2
p
q
+
2
q
2
÷
p
2
−
q
2
6
Rewrite the division as a multiplication
of the first expression times the
reciprocal of the second.
p
3
+
q
3
2
p
2
+
2
p
q
+
2
q
2
·
6
p
2
−
q
2
Factor the numerators and denominators
and then multiply.
(
p
+
q
)
(
p
2
−
p
q
+
q
2
)
6
2
(
p
2
+
p
q
+
q
2
)
(
p
−
q
)
(
p
+
q
)
Simplify by dividing out common factors.
(
p
+
q
)
(
p
2
−
p
q
+
q
2
)
6
3
2
(
p
2
+
p
q
+
q
2
)
(
p
−
q
)
(
p
+
q
)
Simplify.
3
(
p
2
−
p
q
+
q
2
)
(
p
−
q
)
(
p
2
+
p
q
+
q
2
)
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Before doing the next example, let’s look at how we divide a fraction by a whole number. When we divide
3
5
÷
4 , we first write 4 as a fraction so that we can find its reciprocal.
3
5
÷
4
3
5
÷
4
1
3
5
·
1
4
We do the same thing when we divide rational expressions.
Divide:
a
2
−
b
2
3
a
b
÷
(
a
2
+
2
a
b
+
b
2
)
.
Solution
a
2
−
b
2
3
a
b
÷
(
a
2
+
2
a
b
+
b
2
)
Write the second expression as a fraction.
a
2
−
b
2
3
a
b
÷
a
2
+
2
a
b
+
b
2
1
Rewrite the division as the first
expression times the reciprocal of the
second expression.
a
2
−
b
2
3
a
b
·
1
a
2
+
2
a
b
+
b
2
Factor the numerators and the
denominators, and then multiply.
(
a
−
b
)
(
a
+
b
)
·
1
3
a
b
·
(
a
+
b
)
(
a
+
b
)
Simplify by dividing out common factors.
(
a
−
b
)
(
a
+
b
)
3
a
b
·
(
a
+
b
)
(
a
+
b
)
Simplify.
(
a
−
b
)
3
a
b
(
a
+
b
)
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Remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.
