Chapter 8 – 2: NOTES
Simplifying Rational Expressions
Definitions and General Properties of
Rational Numbers and Rational Expressions
a
A rational number can be written as a quotient of two integers, in the form b ,
where the denominator, b, is not 0.
A rational expression is the indicated quotient of two polynomials where
the value of the denominator is assumed to be nonzero.
Sign rules of rational numbers and expressions:
-a a
a
= =1.
b -b b
-a a
=
2.
-b b
Fundamental Principle of Fractions:
If b and k are nonzero integers and a is any integer, then
ak a
= .
bk b
Simplifying a rational expression:
1. Completely factor the polynomial given in the numerator and
2. Apply the fundamental principle of fractions by dividing the common
3. The simplest form will be the quotient of the product of remaining
Warm-up 1. Reduce to lowest terms:
a)
- 80
22225
=
=
- 96 2  2  2  2  2  3
b)
c)
2(m + 3)(
)
2 m 2 - 18
=
=
2
( )(
)
m - 6m + 9
d)
- 12yz
223 y  z 
=20y
225 y
2
)(
)
x -4 (
=
= (-1)
5(2 - x)
10 - 5x
-x-2


= -
 or
5


1
=-
denominator
factor or fac
factors in the
Problems - Simplify:
1.
- 90 a 3 c
- 96 abc 2
2.
9 y 2 + 27y
3
y + 27
3.
ay + 3a - 4y - 12
2ay + 4a - 8y - 16
4.
a - 36
18 - 3a
2
Multiplying and Dividing Rational Expressions
Multiplying Rational Expressions
Basic definition for multiplying rational numbers:
If a, b, c, and d are integers with b and d not equal to zero,
a c a  c ac
=
then  =
.
b d b  d bd
Multiplying rational expressions:
1. Completely factor each numerator and denominator.
2. Apply the basic definition for multiplying rational numbers by
rewriting the numerator as a product of factors and rewrite the
denominator as a product of factors.
3. Simplify by dividing common factors.
4. The result is the quotient of the product of remaining factors in the
numerator and the product of remaining factors in the
denominator.
Warm-up 1. Multiply and simplify:
a)
-5 7
5 7

=
=
14 - 35 2  7  5  7
b)
)(
5ab a 2 - 49 5ab(
 2
=
a + 7 a - 7a
a(a + 7)(
)(
2 a 2 - a - 15 3 a 2 + 8a - 3 (2a + 5)(
 2
c)
=
2
6 a + 17a + 5 (a + 3)(a - 3)(
a -9
)(
)(
2
)
)
=
)
)
= ____
Problems - Perform the indicated operation. Express in simplest form:
1.
6 -7

21 36
2.
3xy 2 x 2 + 20x + 50

x+5
xy - 5y
4 2 - 4n - 24 3 n2 + 8n - 3
3. n 2

3 n + 5n - 2
9 - n2
Dividing Rational Expressions
Basic definition for dividing rational numbers:
If a, b, c, and d are integers with b, c, and d not equal to zero, then
a c a d ad
 =  =
.
b d b c bc
c
d
Note: and are called reciprocals or multiplicative inverses.
d
c
Dividing rational expressions:
1. Apply the basic definition for dividing rational numbers.
2. Follow the steps for multiplying rational expressions in summary 1.
Warm-up 2. Divide and simplify:
a)
c)
4 x 2 24 x 2 y 2 4 x2

= 2
15xy
5 y2
5y
4  15 

=
=
5  24 

5 10 5 - 21

= 
=7 - 21 7 10
b)
3 m2 + 2m - 1 - 27 + 15m - 2 m2 3 m2 + 2m - 1


=
3 m2 + 14m - 5
2 m2 - 7m - 9
3 m2 + 14m - 5
(3m - 1)(m + 1)(
)( )
=
=( )( )(-3 + m)(9 - 2m)
3
Problems - Perform the indicated operation. Express in simplest form:
4.
5 8
7


12 15 18
5.
2
5xy
y - 7y
 2
y + 7 y - 49
6.
xy + xb + cy + cb 18 x3 + 45 x 2 - 27x

xy - 2xb + cy - 2cb
3 x3 - 27x
4
Chapter 8 – 3: NOTES
Adding Rational Expressions
Basic definition for adding rational numbers with a common
denominator:
a c a+c
If a, b, and c are integers and b is not zero, then + =
.
b b
b
Equivalent fractions are fractions with different denominators that name the
same number.
The LCD, least common denominator, is the least common multiple of a set of
denominators. The LCD can be determined by inspection or by writing the
product of the highest power for each factor in any given polynomial.
Adding rational expressions:
1. Determine if the given rational expressions have a common
denominator.
2. If not, find the LCD.
3. Rewrite the given rational expressions as equivalent rational
expressions using the LCD in the denominators.
4. Apply the basic definition for adding rational numbers with a common
denominator.
5. Express result in simplest form.
Warm-up 1. Add and simplify:
a)
2x
4
2x +
2( )
+
=
=
= -------x+2 x+2
x+2
x+2
c)
3
2
+
2x + 1 x - 5
=
3 (
(2x + 1)(
3
x
The LCD is _______.
+
2
2x 5
3( ) x( )
= 2
=
=
+
+
10 x2 10 x2 10 x 2
2 x ( ) 5( )
b)
The LCD is _______________.
)
2 (
+
) (x - 5)(
)
3x +2
7x =
=
+
)
(x - 5)(2x + 1) (x - 5)(2x + 1) (x - 5)(2x + 1)
5
Problems - Add and express in simplest form:
2 3 7
1. - + +
5 20 8
3. 4 +
8
2x - 1
2.
n + 6 2n - 1
+
9
12
4.
4x - 12 x + 1
+
2
x - 9 x+3
Subtracting Rational Expressions
Basic definition for subtracting rational numbers with a common
denominator:
a c a-c
If a, b, and c are integers and b is not zero, then - =
.
b b
b
Subtracting rational expressions:
1. Determine if the given rational expressions have a common
denominator.
2. If not, find the LCD.
3. Rewrite the given rational expressions as equivalent rational
expressions using the LCD in the denominator.
4. Apply the basic definition for subtracting rational numbers with a
common denominator.
5. Express in simplest form.
Warm-up 2. Subtract and simplify:
a)
4(
4m
20
( )-( )
=
=
m-5 m-5
m-5
(
)
)
=
(2x)( ) (5)( )
2x 5
- =
The LCD is __________.
x + 3 x (x + 3)( ) (x)( )
(
) (
)
( )-( )
=
=
=
x(x + 3) x(x + 3)
x(x + 3)
x(x + 3)
b)
6
c)
=
(5x)( ) (2)(
5x
-2=
x+3
(x + 3)( ) (1)(
-
(x + 3) (x + 3)
=
)
)
5x - 2( )
(x + 3)
The LCD is __________.
=
5x - x+3
Problems - Subtract and simplify:
5.
x+1 x - 2
4
6
6.
5 3
7
- 2x 2 x 8x
7.
5
3
2x - 1 x + 3
7
=
More on Rational Expressions and Complex Fractions
Complex fractions are fractional forms that contain rational numbers or
rational expressions in the numerators and/or denominators.
Simplifying a complex fraction - Method A:
1. If necessary, perform the indicated operation in the numerator and/or
denominator.
2. Rewrite the indicated quotient as the division of two rational numbers
or rational expressions.
3. Apply the basic definition for dividing rational numbers.
Simplifying a complex fraction - Method B:
1. Find the LCD of the rational numbers or rational expressions in the
given complex fraction.
2. Multiply both numerator and denominator by the LCD to obtain an
equivalent fraction. (If necessary, apply the distributive property.)
3. Simplify the resulting expression.
Warm-up -
+ 65 (12)
a)
=
- 83 (24)
3
4
5
12
=
3 ( ) 3
+
x
x
x
b)
=
5 ( ) 5
1x
x
x
Simplify using method A:
19 (

12 (
)
)
(
2+
=
=
19 (

12 (
)
)
=
)
x
(
)
=
2x + 3 (

x
(
)
)
=
2x + 3 (

x
(
x
Simplify using method B:
5 2
(
xy y 2
c)
=
6 3
+
(
x y
 5 2
 5 
) - 2 
( )  - (
 xy 
 xy y  =
 6 3 
6 
) + 
( )  + (
 x
 x y 
 2
) 2 
 y  = 5y - ( )
3
6 y2 + ( )
) 
 y
8
)
)
=
2
(
n
+
3
d)
=
1
5+
(
n+3
6-
2 

) 6 
 n+3
1 

) 5 +

n+3

(
 2 
)

 n+3
 1 
)

 n+3 
)(6) - (
=
(
)(5) + (
Problems - Simplify:
14
1. 15x
8
5xy
8 6
+
y y2
2.
4 2
+
x x2
5
2
3. x - 2 x + 2
7
5
2
x - 4 x+2
9
=
6n + (
5n + (
)- 2
)+ 1
=