Writing in Lowest Terms
Write each rational expression in lowest terms.
(a)
Factor. (Step 1)
= Fundamental property (Step 2)
3
5
=
31 x- 42
51 x- 42
3 x- 12
5 x- 20
EXAMPLE 4
We use the fundamental property of rational expressionsto write a rational
expression in lowest terms.
422 CHAPTER 7 Rational Expressions and Applications
Fundamental Property of Rational Expressions
If is a rational expression and if Krepresents any polynomial, where
then the following is true.
PK
QK
P
Q
KZ0,
QP^1 Q Z^02
This property is based on the identity property of multiplication.
Writing in Lowest Terms
Write each rational expression in lowest terms.
EXAMPLE 3
PK
QK
=
P
Q
#K
K
=
P
Q
# 1 = P
Q
(b)
Write as and as
14 k^2
2 k^3
=
2 # 7 #k#k
2 #k#k#k
k^2 k#k k^3 k#k#k.
14 k^2
2 k^3
(a)
Begin by factoring.
30
72
=
2 # 3 # 5
2 # 2 # 2 # 3 # 3
30
72
Group any factors common to the numerator and denominator.
Use the fundamental property.
NOW TRY
14 k^2
2 k^3
=
7
k
30
72
=
5
2 # 2 # 3
=
5
12
14 k^2
2 k^3
=
712 #k#k 2
k 12 #k#k 2
30
72
=
5 # 12 # 32
2 # 2 # 3 # 12 # 32
since the
denominator is 0
for this value.
xZ4,
The given expression is equal to for all values of x, where (since the denom-
inator of the original rational expression is 0 when xis 4).
(^35) xZ 4
NOW TRY
EXERCISE 3
Write the rational expression
in lowest terms.
21 y^5
7 y^2
NOW TRY ANSWER
- 3 y^3
Writing a Rational Expression in Lowest Terms
Step 1 Factorthe numerator and denominator completely.
Step 2 Use the fundamental propertyto divide out any common factors.
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