Multiplying and Dividing
Rational Expressions
Multiplying and Dividing Fractions
Multiply:
a
b
Divide:
w
x
 
c
d
Multiply by the
reciprocal (flip)
 
y
z
w
x
a c
bd
Multiply
Numerators
Multiply
Denominators
 
Remember to Simplify!
z
y
w z
x y
Example 1
Simplify the following expressions:
9
a)
16
5
b)
6
 
4
6
33
44

4
32
  
31
42

3
4
3
4
1
2
    
62
54
  
22
9
 
22
c)
9
20
12
5
6
 10
12
20
22
9
5
6
 
10
1
1
1
1
10
2
4
22
90
1
2

11
45
The techniques we use to simplify a fraction without
variables (Finding the greatest common FACTOR) is the
same we will use to simplify fractions with variables.
Example 2
These are just the values
of x that make the
expression undefined.
x  2   x  7  3 x  8 

Simplify:

2
x3
Half the work is
done. It is
already factored.
Cancel
common
factors
x2
, x  3 or 2
 x  2   x  7  3x  8
 x  3 x  2 
2
 x  2  x  2  x  7  3x  8
 x  3 x  2 
 x  2  x  7  3x  8
x3
Combine the
fractions by
multiplying
Rewrite any
factors if they
are raised to a
power
Example 3
These are just the values
of x that make the
expression undefined.
3 x  15 3 x 2  15 x  18
Simplify:
 2
, x  5, 3, 2, or 5
2
25  x
x  3x  10
Can NOT
cancel since
its not in
factored
form.
by the reciprocal
3x  15 x  3x  10 Multiply
(Flip the fraction)

2
2
25  x 3x  15 x  18
3  x  5
x  5  x  2 

Factor

2
  x  25  3  x 2  5 x  6 
2
3  x  5
 x  5 x  2

  x  5  x  5  3  x  2  x  3
1

x3
Make sure to Factor
Completely
Cancel
common
factors
OR
Example 3
These are just the values
of x that make the
expression undefined.
3 x  15 3 x 2  15 x  18
Simplify:
 2
, x  5, 3, 2, or 5
2
x  3x  10
Can NOT cancel 25  x
since its not in
factored form.
Almost the
same as x – 5
by the reciprocal
3x  15 x  3x  10 Multiply
(Flip the fraction)

2
2
25  x 3x  15 x  18
3  x  5
x  5  x  2 
Factor


 5  x  5  x  3  x 2  5x  6 
2
x  5  x  2 
3  x  5


  x  5  5  x  3  x  2  x  3
Cancel
TRICK: Factor
1
common

out -1 to make
factors
x3
it the same.
Make sure to Factor
Completely