Worksheet 24 (5.1)
Chapter 5 Rational Expressions
5.1 Simplifying Rational Expressions
Summary 1:
Definitions and General Properties of
Rational Numbers and Rational Expressions
A rational number can be written as a quotient of two integers, in the form
where the denominator, b, is not 0.
A rational expression is the indicated quotient of two polynomials where
value of the denominator is assumed to be nonzero.
a
b
,
the
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
97
denominator
factor or fac
factors in the
Worksheet 24 (5.1)
Warm-up 1. Reduce to lowest terms:
a)
- 80
22225
=
- 96 2  2  2  2  2  3
=
b)
- 12yz
223 y  z 
=20y
225 y
=-
c)
2 m2 - 18
2(m + 3)(
)
=
2
( )( )
m - 6m + 9
98
=
2
)(
)
-4 (
d) x
=
5(2 - x)
10 - 5x
= (-1)

= -

Problems - Simplify:
- 90 a 3 c
1.
- 96 abc 2
3.
ay + 3a - 4y - 12
2ay + 4a - 8y - 16
-x-2

 or
5

2.
9 y 2 + 27y
3
y + 27
2
- 36
4. a
18 - 3a
Worksheet 25 (5.2)
5.2 Multiplying and Dividing Rational Expressions
Summary 1:
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.
99
Warm-up 1. Multiply and simplify:
a)
-5 7
5 7

=
14 - 35 2  7  5  7
=
5ab a 2 - 49 5ab(
)(
 2
b)
=
a + 7 a - 7a a(a + 7)(
=
c)
2 a 2 - a - 15 3 a 2 + 8a - 3 (2a + 5)(
)(
 2
=
2
6 a + 17a + 5 (a + 3)(a - 3)(
a -9
)
)
)(
)(
)
)
=
Worksheet 25 (5.2)
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
3.
4 n2 - 4n - 24 3 n2 + 8n - 3

3 n2 + 5n - 2
9 - n2
Summary 2:
Dividing Rational Expressions
100
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.
Worksheet 25 (5.2)
Warm-up 2. Divide and simplify:
a)
5 10 5 - 21

= 
7 - 21 7 10
=-
b)
4 x 2 24 x 2 y 2 4 x2

= 2
15xy
5 y2
5y
4  15 
=
5  24 


=
c)
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)
=Problems - Perform the indicated operation. Express in simplest form:
4.
5 8
7


12 15 18
101
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
Worksheet 26 (5.3)
5.3 Adding and Subtracting Rational Expressions
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.
Summary 1:
Warm-up 1. Add and simplify:
a)
2x
4
2x +
+
=
x+2 x+2
x+2
102
=
2( )
x+2
=
Worksheet 26 (5.3)
5
7
5 (4) 7 (3)
+
=+
12 16
12(4) 16(3)
21
+
=48
=
3
x
c) 2 +
2x 5
b) -
The LCD is _______.
=
=
=
d)
3(
)
2
2x (
10 x2
)
+
+
x(
5(
)
)
10 x2
10 x 2
3
2
+
2x + 1 x - 5
The LCD is _______________.
3 ( )
2 ( )
+
(2x + 1)( ) (x - 5)( )
3x +2
=
+
(x - 5)(2x + 1) (x - 5)(2x + 1)
7x =
(x - 5)(2x + 1)
=
Problems - Add and express in simplest form:
2 3 7
1. - + +
5 20 8
Worksheet 26 (5.3)
12.
n + 6 2n - 1
+
9
12
103
13. 4 +
14.
8
2x - 1
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.
Summary 2:
Worksheet 26 (5.3)
Warm-up 2. Subtract and simplify:
104
a)
b)
4m
20
( )-( )
=
m-5 m-5
m-5
4(
=
(
=
)
)
2x 5
(2x)( ) (5)(
- =
x + 3 x (x + 3)( ) (x)(
)
)
The LCD is __________.
(
) (
)
x(x + 3) x(x + 3)
( )-( )
=
x(x + 3)
=
=
c)
x(x + 3)
5x
(5x)( ) (2)(
-2=
x+3
(x + 3)( ) (1)(
)
)
The LCD is __________.
=
(x + 3) (x + 3)
5x - 2( )
=
(x + 3)
5x - =
x+3
=
Worksheet 26 (5.3)
Problems - Subtract and simplify:
5.
x+1 x - 2
4
6
105
6.
5 3
7
- 2x 2 x 8x
7.
5
3
2x - 1 x + 3
Worksheet 27 (5.4)
5.4 More on Rational Expressions and Complex Fractions
Summary 1:
Complex fractions are fractional forms that contain rational numbers or
rational expressions in the numerators and/or denominators.
106
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 1. Simplify using method A:
( )
3
5
4+6
a) 5 3 = (12)
24
12 - 8
19 ( )
= 
12 ( )
19 ( )
= 
12 ( )
=
3 ( ) 3
+
x
x
x
b)
=
5 ( ) 5
1x
x
x
2+
Worksheet 27 (5.4)
(
=
)
x
(
)
x
2x + 3 (
=

x
(
2x + 3 (
=

x
(
)
)
)
)
=
107
Simplify using method B:
 5 2
5 2
- 2 ( ) - 2 
xy y
 xy y 
c)
=
6 3
 6 3 
+
( ) + 
x y
 x y 
 5 
)  - (
 xy 
=
6 
( )  + (
 x
5y - ( )
= 2
6 y +( )
(
2
) 2 
y 
3
) 
 y
2 

) 6 
 n+3
1 

) 5 +

n+3

2
(
n
+
3
d)
=
1
5+
(
n+3
6-
(
)(6) - (
(
)(5) + (
=
 2 
)

 n+3
 1 
)

 n+3 
Worksheet 27 (5.4)
=
6n + (
5n + (
=
Problems - Simplify:
14
1. 15x
8
5xy
108
)- 2
)+ 1
8 6
+
y y2
2.
4 2
+
x x2
5
2
3. x - 2 x + 2
7
5
2
x - 4 x+2
Worksheet 28 (5.5)
5.5 Dividing Polynomials
Summary 1:
Dividing a polynomial by a monomial:
1. Divide each term of the polynomial by the common monomial divisor.
2. Reduce each fraction to its simplest form.
Dividing a polynomial by a polynomial:
1. Use the conventional long division format and arrange both the
dividend and the divisor in descending order.
2. The first term of the quotient is determined by dividing the first term in
the dividend by the first term of the divisor.
3. Multiply each term of the divisor by the term of the quotient found in
step 2. Position this product so that it can be subtracted from the
dividend.
4. Subtract.
5. Repeat the process beginning with step 2 using the resulting
polynomial from step 4 as the new dividend.
109
Warm-up 1. Divide:
a)
6 a 2 b3 - 8 a3 b + 12ab
= 2- 2+ 2
3 a2
3a 3a 3a
( ) ( )
+
=( ) 3
a
b) x
3
+ x2 - 3x - 3
x +1
x2 - 3 The quotient is ______________.
┌────────────────
x + 1 │ x3 + x2 - 3x - 3
-(
)
- 3x - 3
-(
)
0 The remainder is _____.
Worksheet 28 (5.5)
c) (2 x4 - 3x + 1)  ( x2 - 5)
2x2
+ 10
┌────────────────────────
x2 - 5 │ 2x4 + 0x3 + 0x2 - 3x + 1
-(
)
10x2 - 3x + 1
-(
)
- 3x + 51
= 2 x2 + 10 +
Problems - Divide:
1.
- 72 a3 b5 + 36 ab4 - 27 a2 b2
- 9ab
4
-1
2. x
x+1
110
(
)
2
x -5
3. (5 y 3 - 2 y 2 + y + 1)  ( y 2 - 2)
Worksheet 29 (5.6)
5.6 Fractional Equations
Summary 1:
The restricted values of a given rational expression are those values that
Solvingmake
equations
the denominator
involving equal
rational
to 0.expressions:
1. Determine the restricted values for each rational expression in the
given equation.
2. Find the LCD in the equation.
3. Apply the multiplication property of equality, using the LCD, to clear
the equation of all fractional forms.
4. Solve the resulting equivalent equation.
5. Check when directed to do so.
Note: If a restricted value results as a solution to the equation
found in step 4, then the solution set is .
Warm-up 1. Solve:
a)
5
4 3
= 6
3x 4x
The restricted value is _____, (x  ____).
The LCD is _______.
 4 3 
 5 
(
) 
= ( )

 3x 4x 
 6 
 5 
 4 
 3 
(
) 
)
)
=(
-(

 6 
 3x 
 4x 
=( ) - ( )
10x
=
10x
=
x
111
The solution set is {
}.
Worksheet 29 (5.6)
2
2
3y
=
+
y-2 5
5y - 10
b)
The restricted value is _____, (y  ____).
The LCD is _________.
 2
 3y 
2

(
) 
+  = (
)
 y-2 5
 5(y - 2) 
(
 2 
 + (
) 
 y-2 
10 + (
 2
)
 5
)-(
6 +(
 3y 


) 
 =(

 5(y - 2) 
)
= 3y
) = 3y
y
=
The solution set is ______.
Problems - Solve:
1.
4
3
4
+ = 2
x-1 x x - x
2.
a
3
6
+ =
a -1 2 a -1
Worksheet 29 (5.6)
112
Summary 2:
Ratio and Proportion
A ratio is the comparison of two numbers with division.
A proportion is a statement of equality between two ratios.
The Cross-Multiplication Property of Proportions:
If ba = dc where b0 and d0, then ad = bc .
Solving fractional equations as proportions:
1. Find restricted values for each rational expression in the given
equation.
2. Determine if the given equation is in the form ba = dc .
3. If so, apply the cross-multiplication property of proportions.
4. Solve resulting equivalent equation.
5. Check when directed to do so.
Warm-up 2. Solve:
a)
10
3x + 2
10(
)
20x - 10
2x
x
6
2x - 1
)
= 6(
=( )+(
=
=
=
)
The solution set is {
}.
Problem - Solve:
3.
-5
4
=
x - 1 2x + 1
Worksheet 29 (5.6)
Summary 3:
Word Problems Using Fractional Equations
Follow the listed steps in Summary 3 for Section 2.4.
Warm-up 3. Set up and write an algebraic equation, then solve:
a) The sum of two numbers is 40. If the larger is divided by the
smaller, the quotient is 5 and the remainder is 4. Find the numbers.
113
Declare the variable:
Let
x
= larger number
= smaller number
Write an algebraic equation and solve:
x
(
(
(

)


)

=5+
( )
40 - x
)
x 
4 

) 5+
 =(

40 - x 
40 - x 

x 
 4 
)( 5 )+(
)
 =(

40 - x 
 40 - x 
)
x = 200 - 5x + (
6x = ( )
x =
The numbers are _____ and _____.
Problem - Set up and write an algebraic equation, then solve:
4. The perimeter of a rectangle is 104 meters. If the ratio of its width to its length is 5
to 8, find the dimensions of the rectangle.
Worksheet 30 (5.7)
5.7 More Fractional Equations and Applications
Summary 1:
The restricted values of a given rational expression are those values that
Solvingmake
fractional
the denominator
equations:equal to 0.
1. Factor the denominators.
2. Determine the restricted values for each rational expression in the
given equation.
3. Find the LCD in the equation.
4. Apply the multiplication property of equality, using the LCD, to clear
the equation of all fractional forms.
5. Solve the resulting equivalent equation.
6. Check when directed to do so.
Note: If a restricted value results as a solution to the equation
found in step 5, then the solution set is .
114
3x + 1 5x - 1 x + 3
+
=
2
x - 9 x+3 x - 3
3x + 1
5x - 1 x + 3
+
=
( )( ) x + 3 x - 3
Warm-up 1. Solve:
a)
The restricted values are ___ and ___, (x  ___ and x  ___).
The LCD is _____________.
 3x + 1
5x - 1 

 (
+
 (x + 3) (x - 3) x + 3 
 3x + 1 

 (
 (x + 3) (x - 3) 
 5x - 1 
)+ 
(
 x+3 
(3x + 1) (
 x+3
)= 
(
 x-3 
) + (5x - 1) (
3x + 1 + 5x - 16x + 3 = (
)(
)=0
x = ______ and x = ______
The solution set is {
Problems - Solve:
2
x
4
x + 16
- 2
=
1.
x + 4 x - 16 x - 4
2.
2
4
a
a - 2a + 2
= 2
a + 1 3 - a a - 2a - 3
115
}
)
Worksheet 30 (5.7)
)
)=0
(
)
)
) = (x + 3) (
2
(
 x+3
)= 
(
 x-3 
Worksheet 30 (5.7)
Word Problems Using Fractional Equations
Follow the listed steps in Summary 3 for Section 2.4.
Summary 2:
Warm-up 2. Set up and write an algebraic equation, then solve:
a) A tank can be filled by the hot-water faucet in 4 minutes and by the
cold-water faucet in 6 minutes. How long does it take to fill the
tank using both faucets together?
Declare the variable:
Let
x = time required using both faucets
Write an algebraic equation and solve:
116
1
(
(
 1 
)  +(
 4 
+
1
=
1
) ( ) ( )
 1 
 1
) =(
)
 6 
 x
3x + ( ) = ( )
5x = (
x=



)
The amount of time it takes to fill the tank with both faucets is
______ minutes.
Worksheet 30 (5.7)
Problems - Set up and write an algebraic equation, then solve:
3. John can pick a bushel of peaches in 30 minutes. His little sister can pick a bushel of
peaches in 45 minutes. How long will it take them to pick a bushel of peaches
working together?
4. One pipe can fill a tank in 6 hours. The drain can empty the tank in 10 hours. How
long will it take to fill the tank if the drain is left open?
117
118