2.3 – Multiplying Rational Expressions
The same rules apply to multiplying rational expressions as to when multiply any rational
numbers (i.e. fractions).
a) multiply across (numerator with numerator and denominator with denominator)
b) Reduce to lowest terms by dividing out any common factors
c) State any restrictions. You will need to state restriction on both top and bottom for any
divisor term as it gets flipped during the operation.
Simplify the following expression. State all restrictions.
Example 1:
To avoid large
numbers you
can cancel out
before
multiplying
a)
3x 3
2y2
=
×
b)
10 y 3
9x 2
30 x 3 y 3
18 x 2 y 2
5 xy
=
, x, y ≠ 0
3
d)
Restrictions
are easiest to
get from the
factored line
e)
x 2 − x − 20
÷
2ab 14 a 2 b 2
÷
15c 2
5c
2 ab
15c 2
=
×
5c 14 a 2 b 2
1c
=
, a , b, c ≠ 0
7 ab
x 2 + 9 x + 20
x 2 − 6x
x 2 − 12 x + 36
( x − 5)( x + 4) x 2 − 12 x + 36
=
× 2
( x )( x − 6)
x + 9 x + 20
( x − 5)( x + 4) ( x − 6)( x − 6)
=
×
( x )( x − 6)
( x + 4)( x + 5)
( x − 5)( x − 6)
=
, x ≠ − 5 − 4, 0, 6
x ( x + 5)
x−2
x−2
÷
x − 1 x( x − 1)
x − 2 x ( x − 1)
=
×
x −1
x−2
= x, x ≠ 0,1,2
2.3 – multiplying rational expressions
c)
x−3
x2 + x − 6
×
2
x + 2 x − 15 x − 2
( x + 3)( x − 2) ( x − 3)
=
×
( x + 5)( x − 3) ( x − 2)
x+3
=
, x ≠ −5,2,3
x+5
formally stating restrictions (partial example)
Invert and
multiply
x 2 + 9 x 2 + 20 ≠ 0
( x + 4)( x + 5) ≠ 0
∴ x − 4 ≠ 0 or x + 5 ≠ 0
x≠4
or
x ≠ −5
Restrictions on original
denominator (before it
was flipped) also need to
be considered. Hence x≠0
is also a restriction
Brackets
indicate that
entire package
must be
considered
together.
2.3 – Multiplying Rational Expressions Practice Questions
1. Simplify the following expressions and state any restrictions.
a)
3x3 8 y 3
×
2 y2 9x
b)
− 4x − 8x4
÷
7
7 y3
c)
x−2
3
×
x−2
6
d)
3( x + 2) x + 2
÷
x −1
x −1
e)
6a 3 5a + 15
×
a+3
8a 3
f)
x2 − 4 4 x − 8
÷
x + 3 3x + 9
g)
x 2 + 7 x + 12 x 2 − x − 6
×
x2 + 4x + 4
x2 − 9
h)
2 y − 3 12 y 2 − 19 y + 5
×
3y −1
4 y2 − 9
i)
x 2 + 3 xy
x2 − 9 y 2
÷
x 2 − xy − 42 y 2 x 2 − 10 xy + 21 y 2
j)
x 2 − xy − 6 y 2 x 2 − 8 xy + 15 y 2
÷
x 2 + 2 xy − 8 y 2 x 2 − xy − 20 y 2
2. Write a simplified expression for the area of a triangle given the information and diagram
below.
x-1
A1 = 2x2 + 10x + 12
A2 = 2x2 – 3x + 1
A1
AT = ?
A2
Answers 1. a)
x+2
1
1
15
3( x + 2)
4 x2 y
, x, y ≠ 0 c) , x ≠ 2 d) 3, x ≠ −2,1 e)
, x, y ≠ 0 b)
, a ≠ −3,0 f)
, x ≠ 2,−3
2
3
4
4
2 x3 y 3
4y − 5
1 3
x
x + 2y
x+4
, y ≠ ,± i)
, x ≠ −6 y,±3 y,7 y j)
, x ≠ −4 y,±2 y,3 y,5 y
, x ≠ −2,−3,+3 h)
x+2
2y + 3
3 2
x + 6y
x − 2y
2. a) At=(x+3)(2x-1), x> ½ since measurements need to be positve
g)
2.3 – multiplying rational expressions