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11.4
Simplifying Rational Expressions
Goals p Simplify a rational expression.
p Use rational expressions to find geometric probability.
VOCABULARY
Rational number A rational number is a number that can be
written as the quotient of two integers.
Rational expression A fraction whose numerator,
denominator, or both numerator and denominator are
nonzero polynomials is a rational expression.
Simplified rational expression A rational expression is
simplified if its numerator and denominator have no factors
in common (other than 1).
SIMPLIFYING FRACTIONS
Let a, b, and c be nonzero numbers.
ac
apc
a
bc
bpc
b
Example 1
6x3
3x 12x2
28
4p7
4
Example: 35
5p7
5
Factoring Numerator and Denominator
2p 3pxpxpx
3x (1 4x)
3x (2 x2 )
3x (1 4x)
2 x2
Factor numerator and
denominator.
Divide out common
factor 3x .
Simplified form
1 4x
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231
Example 2
9 x2
x2 4x 3
Recognizing Opposite Factors
( 3 x )(3 x)
Factor numerator and
denominator.
(x 3)( x 1 )
( x 3 )(x 3)
(x 3)( x 1 )
x3
x1
Factor 1 from ( 3 x ).
Divide out common factor x 3
and write simplified form.
Checkpoint Simplify the expression if possible.
4x(x 7)
1. 8x2
x7
2x
3(5 x)
2. 12(x 5)
1
4
GEOMETRIC PROBABILITY
Region B is contained in Region A. An
object tossed onto Region A is equally
likely to land on any point in the region.
The geometric probability that the
object lands in Region B is
Area of Region B
P .
Area of Region A
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A
B
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232
Example 3
Writing and Using a Rational Model
A coin is tossed onto the large rectangular region
shown at the right. It is equally likely to land on
any point in the region.
a. Write a model that gives the probability that
the coin will land in the small rectangle.
b. Evaluate the model when x 4.
2x
3x 9
x
4x
Solution
a. P Area of small rectangle
Formula for geometric probability
Area of large rectangle
x( 2x )
Find areas.
4x( 3x 9 )
2p x p x
4x p 3(x 3)
Divide out common factors.
x
6(x 3)
Simplified form
b. To find the probability when x 4, substitute 4 for x in the model.
P
x
6( x 3 )
4
6( 4 3 )
4
2
42
21
2
Answer The probability of landing in the small rectangle is .
21
Checkpoint Refer to Example 3.
3. Find the probability when x 5.
5
48
Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved.
Chapter 11 • Notetaking Guide
233