PERIOD
DATE
NAME
11-3 Study Guide and Intervention
Simplifymg
Rational Expressions
Identify Excluded Values
Rational
Expression
an algebraic fraction with numerator and denominator that
are polynomials
Example:
Because a rational expression involves division, the denominator cannot equal zero. A n y value o f the denominator that
r e s u l t s i n d i v i s i o n b y z e r o i s c a l l e d a n excluded value o f t h e d e n o m i n a t o r .
Example 1: State the exetederf value of
4m-8
7n
X
Exclude the values for w h i c h m + 2 = 0.
/w + 2 = 0
/w + 2 — 2 = 0 - 2
m = -2
+1
Example 2 : State the excluded values of ^77^.
+ 2•
Exclude the values for w h i c h
The denominator cannot equal 0 .
The denominator cannot equal 0 .
- 9=0
Subtract 2from each side.
Simplify.
Therefore, m cannot equal - 2 .
- 9 = 0.
(x + 3)ix - 3 ) = 0
Factor.
X + 3 = 0 or X - 3 = &
Zero Product Property
= -3
= 3
Therefore, x cannot equal - 3 o r 3.
Exercises
State the excluded values for each rational expression.
1.
2b
2.
4.
3.
2TI-
5. jl2
7.
9.
11.
13.
12
_4
6.
8.
a:^ + 4x + 4
fc^-2fe+l
10.
fe2 + 4fc + 3
25-n^
n2 - 4n
12.
- 5
14.
+ 4n - 5
fc^ + 2 t e - 3
Chapter 11
32
+ a
-4
2m^-8
Zx + 18
- 16
a-l
5a +
a?+
6
mz-l
2771^ - m -
2x^
+ 5x +
1
1
x ^ - W x + \
-2n-3
15. fc2 - 2 0 k
1 2 - a
y ^ - y - 2
3y2 - 1 2
X
• +
4x + 4
16. 4x2 + i i j . _ 3
+ 64
17
Glencoe Algebra 1
NAME
PERIOD
DATE
11-3 Study Guide and Intervention fconf/nuedj
Simplifying
Rational Expressions
Simplify Expressions F a c t o r i n g p o l y n o m i a l s i s a u s e f u l t o o l f o r s i m p l i f y i n g r a t i o n a l e x p r e s s i o n s . T o s i m p l i f y a r a t i o n a l
expression, first factor the n u m e r a t o r and denominator. T h e n divide each b y the greatest c o m m o n factor.
54^3
Example 1; Simplify
54z_ _ (6zX9z )
24yz
numerator and the denominator is 6 z .
(6z)(4y)
I
Divide the numerator and denominator by 6z.
.^^^^^^
^(^)(4y)
9z2
Simplify.
4y
3x —
9
Example 2: Simplify ^2 _ 5;,. + ^' State the excluded values of je.
^
3j>-^^
DividebytheGCF,x-3.
=
Simplify.
Exclude the values for w h i c h
x^-5x
-5x + 6 = Q.
+ 6=0
(x-2)(x-3) =0
=2 or
X
X=
3
T h e r e f o r e , x 7^ 2 a n d x
3.
Exercises
Simplify each expression. State the excluded values of the variables.
1.
,
3.
12ab
. 7n^
2. •
x+2
-r;—r
4. •
„,„
2TI-8
^' n2 -
16
7
-4
* x2 + 4x + 4
9
fc^
- 1
^^-25
+ 2X +
,
1
x2 - 1
g
+3a + 2
'02 + 50 + 6
jQ Tn2
* k2 + 4fe + 3
11
m2-4
.
- 2771
+ 1
• 27n2 - 771 - 1
_
+3:-6
^^'712-471-5
2x2-8
n2 + 7 n + 12
y2-y-2
^•^'712+277-8
Chapter 11
18
Glencoe Algebra 1
NAME
DATE
PERIOD
11-4 Study Guide and Intervention
Multiplying a n d Dividing Rational Expressions
Multiply Rational Expressions T o m u l t i p l y r a t i o n a l e x p r e s s i o n s , y o u m u l t i p l y t h e n u m e r a t o r s a n d m u l t i p l y t h e
denominators. Then simplify.
Example 1: Find
5ab2
2cV
a^b _ 2 a H c ^ f
5ab^
3c/
3c/*
Multiply.
ISab^cf
}(abefX2ac)
^
Simplify.
Kai>£f){i5b)
2ac
Simplify.
15b
Example 2: Find
-16
2x + 8
x + 4
2x + 8
x^ + Sx +16'
x+4
_ { x - 4)(x + 4)
x^ + S x + i e
2(x + 43
X +4
(x + 4)Cx + 4)
Factor.
Simplify.
(a: + 4 ) ^ » - F ^ i
X ~ 4
Multiply.
2X + 8
Exercises
Find each product.
1.
3.
5.
7.
9.
11.
13.
6ab
2.
x +2
jc - 4
X - 4
2 7 1 - 8
271 +
8x +
6.
x - 1
8
- 2 X +1
x^ + 6 X + 8
+ 9 X+4
2x^
4
71-4
71 + 2
X ^
4.
X - 1
2 x ^ - x - l
x ^ - 3 x +2
7l2
7l2 -
771 +
10
+ 7 a + 12
a2+2a-8
Chapter 11
8.
2X + 2
- 25
7 l 2 + 671
+5
+ 3 a - 10
a2+2a-8
mp
771-5
16
8
771-5
x ^ - 6 4
x+8
2x + 1 6
x^ + 16x + 64
a^ - 2 5
10.
12.
14.
-4
a +2
a - 5
771^-1
2771^ -
2771
771 -
1
3p - 3r
lOpr
+1
2771 +
1
20p^r^
p2 _ J.2
v"^-417-21
3p2 + 6v
23
771^ -
v^ + 8v
+ 1117 + 24
Glencoe
Algebra 1
NAME
PERIOD
DATE
11-4 Study Guide and Intervention
(continued)
Multiplying and Dividing Rational Expressions
Divide Rational Expressions T o d i v i d e r a t i o n a l e x p r e s s i o n s , m u l t i p l y b y t h e r e c i p r o c a l o f t h e d i v i s o r . T h e n s i m p l i f y .
Findg^.^.
Example 1:
IZe^f
. c^f^ _ 1 2 e ^ f
lOab
X
2tqW
24
abf
Example 2: Find f ^^^"^
x^ +
6x - 27
x-3
x2 + X - 2
x2 + l l x + 18
- ."^"^
+ 6x - 27
_
x2 + l l x + 18
( X + 9 ) ( X - 3)
(x + 9 ) ( X
+
2)
x^ +
X
- 2
x-3
^
(X +
2)(x - 1)
x-3
^(x^4-2-)(x - 1)
=
X - l
Exercises
Find each quotient.
^
12ab . b
,
3xy2
^
.
-Sxy
777-
5
771
8
2n-4
y2 - 49 • y + 7
71
_ x2-5x + 6 . x - 3
5
11.
13.
16
.71^-4
2n
9.
-5
4.
'
x^ +
6x + 8
x2 +
4X
.
+ 4
7 1 ^ - 571 +
X +
4
X +
2
6
7l2 + 371
a^ + 7 a
+ l2 .
a2 + 3a
- 10
Chapter 11
_ a^b^c . 6a^bc
15
3
471
-71
+ 12
- 25
10.
12.
7 7 1 ^ - 4 9 . 7 7 1 ^• -13771 + 42
377l2
771
p2-2pr + r2 .
p + r
-9
14.
2a2 +
13a-7 "
24
p +r
a +3
4a2
-
1
Glencoe Algebra 1