SIMPLIFYING EXPRESSIONS
11.1.1
To simplify rational expressions, determine factors in the numerator and denominator
that are the same and then write them as fractions equal to 1. For example:
6
=1
6
x+2
=1
x+2
x2
=1
x2
3x − 2
=1
3x − 2
The rational expressions below cannot be simplified to equal 1 (or anything else):
6+5
6
x3 + y
x3
3x − 2
2
x
x+2
As shown in the examples below, most problems that involve simplifying rational
expressions will require factoring the numerator and denominator.
Note that in all cases it is assumed that the denominator does not equal zero, so in
Example 4 below the simplification is only valid provided x ≠ –6 or 2. For additional
information, see examples 1 and 2 in the Math Notes box in Lesson 11.1.3.
One other special situation is shown in the following examples:
− ( x + 2)
− ( x − 5)
−2
−x
−x − 2
5−x
= −1
= −1
⇒
⇒ −1
⇒
⇒ −1
2
x
x+2
x+2
x−5
x−5
Again, it is assumed that the denominator does not equal zero.
Example 1
Example 2
2
3
12
2⋅2⋅3
2
= =1
⇒
⇒ since
54
2 ⋅ 3⋅ 3⋅ 3 9
2
3
6x 3 y 2
2 ⋅ 3⋅ x 2 ⋅ x ⋅ y 2
2x
⇒
⇒
15x 2 y 4
5 ⋅ 3⋅ x 2 ⋅ y 2 ⋅ y 2
5y 2
Example 3
Example 4
12 ( x − 1)
x 2 − 6x + 8
( x − 4)( x − 2)
⇒
2
x + 4x − 12
( x + 6)( x − 2)
( x + 2 ) ⇒ 4 ⋅ 3 ( x − 1 )2 ( x − 1 ) ( x + 2 )
2
2
2
3 ( x − 1) ( x + 2 )
3 ( x − 1) ( x + 2 ) ( x + 2 )
3
3
4(x − 1)
⇒
since ,
(x + 2)
3
122
( x − 1)2 , and
( x − 1 )2
x+2
=1
x+2
⇒
© 2015 CPM Educational Program. All rights reserved.
(x − 4)
( x + 6)
since
( x − 2) = 1
( x − 2)
Core Connections Integrated III
Chapter 11
Problems
Simplify each of the following expressions completely. Assume the denominator does not equal
zero.
1.
2 ( x + 3)
4 ( x − 2)
2.
2 ( x − 3)
6 ( x + 2)
3.
2 ( x + 3) ( x − 2 )
6 ( x − 2 )( x + 2 )
4.
4 ( x − 3) ( x − 5 )
6 ( x − 3) ( x + 2 )
5.
3( x − 3) ( 4 − x )
15 ( x + 3) ( x − 4 )
6.
15 ( x − 1) ( 7 − x )
25 ( x + 1) ( x − 7 )
7.
24 ( y − 4 ) ( y − 6 )
16 ( y + 6 ) ( 6 − y )
8.
36 ( y + 4 ) ( y − 16 )
32 ( y + 16 ) (16 − y )
9.
( x + 3 )2 ( x − 2 ) 4
( x + 3 )4 ( x − 2 )3
10.
( 5 − x )2 ( x − 2 )2
( x + 5 )4 ( x − 2 )3
11.
( 5 − x )4 ( 3x − 1)2
( x − 5 )4 ( 3x − 2 )3
12.
13.
x 2 + 5x + 6
x2 + x − 6
14.
2x 2 + x − 3
x 2 + 4x − 5
15.
17.
x2 − 1
( x + 1) ( x − 2 )
18.
20.
x 2 − 16
x 3 + 9x 2 + 20x
21.
16.
19.
24 ( 3x − 7 ) ( x + 1)
20 ( 3x − 7 )
3
6
( x + 1 )5
x2 − 4
x2 + x − 6
12 ( x − 7 ) ( x + 2 )
20 ( x − 7 )
2
4
( x + 2 )5
x 2 + 4x
2x + 8
x2 − 4
( x + 1 )2 ( x − 2 )
2x 2 − x − 10
3x 2 + 7x + 2
Answers
1.
4.
7.
10.
13.
16.
19.
( x + 3)
2 ( x − 2)
2 ( x − 5)
3( x + 2 )
3( y − 4 )
−
2 ( y + 6)
( 5 − x )2
( x + 5 )4 ( x − 2 )
x+2
x−2
6 ( x + 1)
5 ( 3x − 7 )
2.
5.
8.
11.
14.
2
x+2
x+3
Parent Guide and Extra Practice
17.
20.
( x − 3)
3( x + 2 )
( x − 3)
−
5 ( x + 3)
9(y + 4)
−
8 ( y + 16 )
( 3x − 1)2
( 3x − 2 )3
2x + 3
x+5
x −1
x−2
x−4
x(x + 5)
3.
6.
9.
( x + 3)
3( x + 2 )
3( x − 1)
−
5 ( x + 1)
( x − 2)
( x + 3 )2
12.
3
5 ( x − 7 )( x + 2 )
15.
x
2
18.
21.
x+2
( x + 1 )2
2x − 5
3x + 1
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