Download B.2 Rational Algebraic Expressions

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B.2
Rational Expressions
Reduction of rational expressions
Algebra of rational expressions
Complex rational expressions
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.
Definition of Rational Expression
A rational expression in x is a ration
R x  
p x 
q x 
Where p( x ) and q ( x ) are polynomials in
q  x  0
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x and
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E - BOOK FOR COLLEGE ALGEBRA
Note:
Since an integer is a polynomial of degree
zero, every rational number is a rational
expression. Other examples of rational
expressions are
x  x 1
R x  
2x  5
2
2a 2  5a  6
W a   3
a  8a  1
y 2  5y  6
Q y  
y 2 1
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Definition:
p  x
The domain of the rational function R  x  
q  x
consists of all real numbers except those for
which q (x ) = 0
Definition
p  x
The value of the rational expression R  x  
q  x
when x is equal to a is obtained by replacing x
by a in R ( x ).
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E - BOOK FOR COLLEGE ALGEBRA
Example 1
3x  2
1
find R  
R  x 
2
2x 1
Solution
1
3   2
2
1
R   
 2  2 1  1
 
2
7
3
2
 2
 2
2
11
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7

4
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E - BOOK FOR COLLEGE ALGEBRA
Example 2
Find the domain of
3x 2  2 x  1
Q  x 
x2 1
Solution
3x 2  2x  1 3x 2  2x  1
Q x  

2
x 1
(x  1)(x  1)
The domain of Q( x ) is all real numbers except
x = 1 and -1
(, 1)  (1,1)  (1, )
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E - BOOK FOR COLLEGE ALGEBRA
Reducing Rational Expressions
p  x
Reducing R  x  
q  x
consists of two steps:
Step 1 Factor both p  x  and q  x 
Step 2 Divide both p  x  and q  x  by the greatest
common factor.
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King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Example 3
3x  9x  6
1.
2
2x  8
2
Reduce to lowest terms
3  x  3x  2 
2



2  x 2  4
3  x  2  x  1
2  x  2  x  2 
3  x  1
2  x  2
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, x  2
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3w  3w
2.
w 3 w
4


3w w 3  1
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w w 2  1
3w w  1 w 2  w  1
w w  1w  1
3 w  w  1
2

w 1
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,w  0,1, 1
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E - BOOK FOR COLLEGE ALGEBRA
Multiplication and Division of
Rational Expressions
p x  r x  p x  r x 


Multiplication:
q  x  t  x  q  x t  x 
Division:
p x 
r x 
p  x  t  x  p  x t  x 




q x  t x  q x  r x  q x  r x 
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E - BOOK FOR COLLEGE ALGEBRA
Example 4
Perform the indicated operations
and reduce the resulting rational
expression.
6x  8
4x  4x  1
1.

2
2x  5x  3
6x  3
2
2  3x  4  2x  1

 2x  1 x  3 3  2x  1
2
2  3x  4 
1

, x 
3  x  3
2
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9  6x  x
x  xk  3x  3k
2.

2
9  6x  x
x 2 9
2

2
2
2
9

6
x

x
x

  9
2
2
9

6
x

x
x

  xk  3x  3k 
x  3  x  3 x  3


2
 x  3 x  x  k   39x  k 
2
x  3  x  3


2
 x  3  x  3 x  k 
3
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x  3


,x
 x  3 x  k 
2
 3, 3,  k
King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Addition and Subtraction of
Rational Expressions
Identical denominators
a c a c
 
b b
b
Different denominator
Least Common Denominator(L.C.D)
a c ad  bc
 
b d
bd
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Example 5
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Find the L.C.D.
x 2
2x
and
2
x  2x  15
x 2  4x  5
Solution
d1  x 2  2x 15   x  3 x  5
d 2  x 2  4x  5   x  5 x  1
L .C .D .   x  5 x  1 x  3
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E - BOOK FOR COLLEGE ALGEBRA
Example 6
Perform the indicated operations and
simplify.
4x  a 3x  1
1. 2

2
x a
ax
Solution
The L.C.D.= (x – a) ( x + a)
1 3x  1 x  a 

4x  a


 x  a  x  a 
 x  a  x  a 
3x  3xa  3x


 x  a  x  a 
2
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3  x 2  xa  x 
 x  a  x  a 
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Example 6. (cont.)
4
2
2
2.

 2
3x x  1 x  x
Solution
The L.C.D.= 3 x ( x + 1)
4  x  1
6x
6



3x  x  1 3x  x  1 3x  x  1
10  x  1
1
10x  10



3x  x  1 3x
3x  x  1
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Complex Rational Expressions:
An algebraic expression which has rational
expressions in its numerator or denominator is
called a complex rational expression. For
example,
1
2

2 3 , and
1 3

4 8
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1
2

x 2 x 2
3
4

x 2
x 2
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E - BOOK FOR COLLEGE ALGEBRA
Complex Rational Expressions:
To simplify such a complex expression we multiply
both its numerator and denominator by the L.C.D.
of all the algebraic expressions involved in the
makeup of the complex expression.
Example 7
Simplify the following complex rational expressions:
a)
2
2

3
3x
2
1

3x 2
6
b)
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a
1

x 2  a2
x a
1
3
 2
ax
a x 2
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Solution
a)
2
2
 2 2 



3
3x  3 3x  6x
 2

2
1
1


  6x
2
2
3x
6
6
 3x
2x 2  4x

4x 2

 x  2
2  x 2  x 
2x
2x

x 2
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E - BOOK FOR COLLEGE ALGEBRA
Solution
a
1

2
2
x a
b) x a
1
3
 2
ax
a x 2
L .C .D   x  a  x  a 
1 
 a

 x 2  a2 x  a 


1
3


 2
2
 ax a x 

a  x  a
x  a  3

 x a  x a 
 x a  x a 
2a  x
33  x  a
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