Haberman / Kling
MTH 95
Section III: Rational Expressions, Equations, and Functions
Module 2: Multiplying and Dividing Rational Expressions
MULTIPLYING RATIONAL EXPRESSIONS
Multiplying rational expressions works the same way as multiplying fractions.
EXAMPLE: Multiplying Fractions
16 15 16 15


45 8
45  8
2  8  15

3  15  8
2

3
(multiply the numerators and the denominators)
(cancel factors that appear in both the
numerator and the denominator)
To multiply two rational expressions, multiply numerators and multiply
denominators:
A C
AC
 
B D
BD
It is often useful to factor the components of the rational expression so
that we can cancel all common factors in the numerator and denominator.
EXAMPLE: Perform the indicated multiplication; simplify your result completely.
a.
25a 6b5

9b8 5a 2
b.
x  1 3x  9

x  3 4 x2  4
2
SOLUTIONS:
a.
25a 6b5 150ab5


9b8 5a 2 45b8 a 2

(multiply the numerators and the denominators)
10  15 a b5
(cancel factors that appear in both the
numerator and the denominator)
3  15 b5 b3 a a
10

3ab3
b.
x  1 3x  9
( x  1)  3( x  3)
 2

x  3 4 x  4 ( x  3)  4( x 2  1)


(multiply the numerators and the denominators)
3 ( x  1) ( x  3)
4 ( x  3) ( x  1) ( x  1)
(cancel factors that appear in both the
numerator and the denominator)
3
, for x  3, x  1
4( x  1)
Try these yourself and then check your answers.
Perform the indicated multiplication; simplify your result completely.
a.
16m2 9n10

12n8 4m5
b.
p 2  10 p  9 3 p  6

p9
p2  4
SOLUTIONS:
a.
16m 2 9n10 16  9m 2 n10


12n8 4m5
12  4n8 m5


(multiply the numerators and the denominators)
4  4  3  3 m 2 n8 n 2
344 n m m
8
3n 2
, for n  0
m3
2
3
(cancel factors that appear in both the
numerator and the denominator)
3
b.
p 2  10 p  9 3 p  6 ( p  1)( p  9) 3( p  2)



p9
( p  2)( p  2) p  9
p2  4


( p  1) ( p  9)  3 ( p  2)
( p  2) ( p  2)  ( p  9)
(factor numerators and denominators)
(cancel factors that appear in both the
numerator and the denominator)
3( p  1)
, for p  2, p  9
p2
DIVIDING RATIONAL EXPRESSIONS
Dividing rational expressions works the same way as dividing fractions.
EXAMPLE: Dividing Fractions
10 20 10 27



9 27
9 20
10  27

9  20
10  9  3

9  10  2
3

2
(multiply by the reciprocal of the divisor)
(factor to facilitate canceling)
To divide two rational expressions, multiply by the reciprocal of the divisor
(i.e., change division into multiplication by “flipping” the second fraction):
A C
A D

 
B D
B C
EXAMPLE: Perform the indicated division; simplify your result completely.
a.
p4
p3
 2
p6
p  6p
b.
x2  x
8x2

4 x  12
x2  9
4
SOLUTIONS:
a.
p ( p  6)
p4
p3
p4
 2


p6
p

6
p  6p
p3


p 4  p ( p  6)
( p  6)  p
3
(cancel factors that appear in both the
numerator and the denominator)
p5
p3
 p 5  3 , for p  0
 p
b.
(multiply by the reciprocal of the divisor)
(use rule of exponents)
2
x2  x
8x2
8 x 2 4 x  12



4 x  12
x2  9
x2  9 x2  x


(multiply by the reciprocal of the divisor)
8 x  x  4 ( x  3)
( x  3) ( x  3)  x ( x  1)
(cancel factors that appear in both the
numerator and the denominator)
32 x
, for x  3
( x  3)( x  1)
Try these yourself and then check your answers.
Perform the indicated division; simplify your result completely.
a.
10
6 w2
 9w
2 w  10
w5
t2  9 t2  t  6
b.

4t  4
2t  2
SOLUTIONS:
a.
10
2
w5
6 w2
 9w  6w  10
2 w  10
w  5 2 w  10 9 w


(multiply by the reciprcal of the divisor)
2  3 w2  ( w  5)
2 ( w  5)  3  3 w2 w8
1
, for w  5
3w8
(cancel factors that appear in both the
numerator and the denominator)
5
t 2  9 t 2  t  6 t 2  9 2t  2
b.



4t  4
2t  2
4t  4 t 2  t  6



(multiply by the reciprcal of the divisor)
(t  3)(t  3)
2(t  1)

2  2(t  1) (t  3)(t  2)
(t  3) (t  3)  2 (t  1)
2  2 (t  1)  (t  3) (t  2)
t 3
, for t  3, t  1
2(t  2)
(factor expressions)
(cancel factors that appear in
numerator and denominator)