MULTIPLYING RATIONAL EXPRESSIONS
Let a, b, c, and d be nonzero real numbers or variable expressions. Then the following property applies:
a·c
a c
· =
, where b 6= 0 and d 6= 0.
b d
b·d
EXAMPLE 1
Original Problem:
18x2
10y 2
·
15y 3
24x
Since neither the numerator (top) nor the denominator (bottom) have addition or
subtraction, we are able to multiply and simplify this rational expression directly.
18x2 15y 3
·
10y 2 24x
Original Rational Expressions
18x2 · 15y 3
270x2 y 3
=
10y 2 · 24x
240xy 2
Multiply the Rational Expressions
270x2 y 3
270 x2 y 3
=
·
·
240xy 2
240 x y 2
Separate the rational expression into independent
fractions using like-terms.
270 ÷ 30
9
=
240 ÷ 30
8
Simplify the fraction.
x2 x·x
x
=
=
x
1
x
Simplify the fraction.
y3
y
y·y·y
=
= 2
y · y
y
1
Simplify the fraction.
270 x2 y 3
9 x y
·
·
= · ·
240 x y 2
8 1 1
Rewrite using simplified fractions.
9·x·y
9xy
=
8·1·1
8
Combine the fractions for solution.
1
EXAMPLE 2
Original Problem:
n2 + 10n + 16
5n − 10
·
n−2
n2
+ 9n + 8
Since the numerators and denominators have addition and subtraction, we need to take
out the GCF and factor them (if possible) before we multiply the rational expressions.
n2 + 10n + 16
Original Numerator 1st Fraction
GCF = 1
No need to remove the GCF!
(n + 8)(n + 2)
Factor the trinomial.
5n − 10
Original Denominator 1st Fraction
GCF = 5
5(n − 2)
Simplify the denominator by removing the GCF.
n−2
Original Numerator 2nd Fraction
GCF = 1
No need to remove the GCF!
n−2
Largest power is not equal to 2, so no factoring!
n2 + 9n + 8
Original Denominator 2nd Fraction
GCF = 1
No need to remove the GCF!
(n + 8)(n + 1)
Factor the trinomial.
Rewrite the rational expression using the factored numerator and denominator, then
multiply.
(n − 2)
(n + 8)(n + 2)
·
5(n − 2)
(n + 8)(n + 1)
New Rational Expressions
(n + 8)(n + 2)(n − 2)
5(n − 2)(n + 8)(n + 1)
Multiply the rational expressions.
Before we can simplify the rational expression, we must identify the n-values that will
make the denominator equal zero. Use the Zero Product Property, from chapter 8, to
2
discover what n-values will make the fraction undefined.
ZERO PRODUCT PROPERTY
If a and b are two numbers where a · b = 0, then a = 0 or b = 0.
n−2=0
+2 + 2
n+8=0
−8 − 8
n+1=0
−1 − 1
n=2
n = −8
n = −1
Since n = 2, n = −8, and n = −1 will make the fraction undefined, we must write n 6= −8, −1, 2.
Now that we have identified which n-values will make the fraction undefined, we are
able to simplify the rational expression.
+ 2)(n
(n + 8)(n + 2)(n − 2)
(n
+8)(n
n+2
−2)
=
=
5(n − 2)(n + 8)(n + 1)
5
(n − 2)
(n + 8)(n + 1)
5(n + 1)
3