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Appendix A
Basic Algebra
Review
Section A-4
Operations on
Rational Expressions
Operations on Rational Expressions




Reducing to Lowest Terms
Multiplication and Division
Addition and Subtraction
Compound Fractions
Barnett/Ziegler/Byleen College Mathematics 12e
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Rational Expressions
A quotient of two algebraic expressions (division by 0
excluded) is called a fractional expression.
If both the numerator and the denominator are polynomials,
the fractional expression is called a rational expression.
1
x 3  2x
5
x
x7
3x 2  5x  1
Barnett/Ziegler/Byleen College Mathematics 12e
x 2  2x  4
1
3
Rational Expressions
AGREEMENT Variable Restriction
Even though not always explicitly stated, we always assume
that variables are restricted so that division by 0 is excluded.
For example, given the rational expression
2x  5
x x  2 x  3
the variable x is understood to be restricted from being 0, –2,
or 3, since these values would cause the denominator to be 0.
Barnett/Ziegler/Byleen College Mathematics 12e
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Reducing to Lowest Terms
Fundamental Property of Fractions
If a, b, and k are real numbers with b, k  0, then
For example, given the rational expression
ka a

kb b
the variable x is understood to be restricted from being 0, –2,
or 3, since these values would cause the denominator to be 0.
Barnett/Ziegler/Byleen College Mathematics 12e
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Reducing to Lowest Terms
Using this property from left to right to eliminate all common
factors from the numerator and the denominator of a given
fraction is referred to as reducing a fraction to lowest terms.
Using the property from right to left–that is, multiplying the
numerator and denominator by the same nonzero factor–is
referred to as raising a fraction to higher terms.
Reduce to lowest terms.

x x  2  x 2  2x  4
x 4  8x

3
2
3x  2x  8x
x x  2 3x  4 
Barnett/Ziegler/Byleen College Mathematics 12e
x 2  2x  4

3x  4

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Multiplication and Division
Multiplication and Division
If a, b, c, and d are real numbers, then
a c ac
1.
 
,
b, d  0
b d bd
a c a d
2.
   ,
b, c, d  0
b d b c
Barnett/Ziegler/Byleen College Mathematics 12e
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Multiplication and Division
Perform the indicated operation and reduce to lowest terms.
10x 3 y
4 x 2  12x
10x 3 y
x2  9


 2
2
3xy  9y
x 9
3xy  9y 4 x  12x
10x 3 y x  3x  3


3y x  3 4x x  3
5 x2
3
11
31
21
x  3 x  3


3 y x  3
4 x x  3
10x y
5x 2

6
Barnett/Ziegler/Byleen College Mathematics 12e
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Addition and Subtraction
Addition and Subtraction
For a, b, and c real numbers,
a c ac
1.
 
,
b b
b
a c ac
2.
 
,
b b
b
Barnett/Ziegler/Byleen College Mathematics 12e
b0
b0
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Addition and Subtraction
Least Common Denominator
The least common denominator (LCD) of two or more
rational expressions is found as follows:
1. Factor each denominator completely, including integer
factors.
2. Identify each different factor from all the denominators.
3. Form a product using each different factor to the highest
power that occurs in any one denominator. This product is
the LCD.
Barnett/Ziegler/Byleen College Mathematics 12e
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Addition and Subtraction
Combine into a single fraction and reduce to lowest terms.
1
1
2
  2
x 1 x x 1
LCD = x(x – 1)(x + 1)
1
1
2

 
x  1 x x  1x  1
x x  1  x  1x  1  2x

x x  1x  1
x 2  x  x 2  1  2x
1 x


x x  1x  1
x x  1x  1

 x  1
x x  1 x  1
Barnett/Ziegler/Byleen College Mathematics 12e

1
x x  1
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Compound Fractions
A fractional expression with fractions in its numerator,
denominator, or both is called a compound fraction. It is
often necessary to represent a compound fraction as a simple
fraction–that is (in all cases we will consider), as the
quotient of two polynomials.
We will use the two different methods.
Barnett/Ziegler/Byleen College Mathematics 12e
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Compound Fractions
Express as a simple fraction reduced to lowest terms. Use
division of rational forms.
1
1

5h 5   1  1 h
 5  h 5  1
h
55h 1


5 5  h  h
h

5 5  h h
1

5 5  h 
Barnett/Ziegler/Byleen College Mathematics 12e
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Compound Fractions
Express as a simple fraction reduced to lowest terms. Multiply
the numerator and denominator by the LCD of all fractions.
y

2
x
y

x
x
2 2 y
x y  2
2
x
y

x
2 2 y
x y  
y
x
x
2 2 y
2 2
x
y

x
y
2
2
y 
x

x
2 2 y
x y  x 2 y2
x
y 
y x
 3

3
xy  x y
3
3
y 2  xy  x 2

xy y  x 
Barnett/Ziegler/Byleen College Mathematics 12e
x
y2
x
y
y  x  y 2  xy  x 2 
xy y  x  y  x 
or
x 2  xy  y 2
xy x  y 
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