Algebraic Fractions
Objectives:
1) Simplify an algebraic fraction.
2) Multiply and divide algebraic fractions
3) Add and subtract algebraic fractions
4) Simplify complex fractions
Terms:
• If x and y are real numbers, y  0, then x/y
is called a fraction.
• x is the numerator
• y is the denominator
• Algebraic fractions are fractions
containing algebraic expressions.
• If the algebraic expressions are polynomials
then the fraction is called a rational
expression.
2
Two fractions are equal if their
cross products are equal.
2a

3
Is
4a
because 2a(6) = 3(4a)
6
3x
4y
2
2

12 y
16 x
2
2
?
No, because(3x2)(16x2)  (4y2)(12y2)
3
To Simplify Fractions divide out all factors that are common to
both numerator and denominator
25  x
Ex.1
x
Ex.2
2
2
 10 x  25
( 5  x )( 5  x )

( x  5 )( x  5 )

5 x
x  5
3
x 8
x
2
 ax  2 x  2 a
( x  2 )( x
2

 2 x  2)

x(x  a)  2(x  a)
( x  2 )( x
2
 2 x  2)
( x  2 )( x  a )

x
2
 2x  2
x  a
4
Multiplying and Dividing
Fractions
• Multiply: numerator x numerator
denominator x denominator
• Divide out common factors
• Divide: Multiply by the reciprocal of the
divisor
5
Multiplying Algebraic Fractions
x
2x
2
2
 x

 3x
x
x ( x  1)
x (2 x  3)
Solution
All factors
2x
2

 x  3
2
1

( 2 x  3 )( x  1 )
( x  1 )( x  1 )
1
:1
reduce
to 1.
6
Dividing Algebraic FractionsWrite the reciprocal and multiply.
ax  bx  a  b
a
2
 2 ab  b
2
ax  bx  a  b
a
a
2
 2 ab  b
 b ( x  1 )
(a  b )
2
2

x

x

x
2
2
2
1
 2x 1
 2x 1
x
2
1
( x  1)


2
( x  1 )( x  1 )

( x  1)
(a  b )
7
Adding and Subtracting
Algebraic Fractions
• Factor the denominator, if necessary
• Find the LCD (least common denominator)
• The LCD can be found by using each factor
the greatest number of times that it appears
in any one denominator.
• Write each fraction as an equivalent fraction
with this LCD.
8
Adding and Subtracting
with unlike Denominators
2
y
2
 3
1
1
y 1
2
( y  1 )( y  1 )
2
( y  1 )( y  1 )
2  3y
2
3



1

2

 1)
1 ( y  1 )( y  1 )
( y  1 )( y  1 )
( y  1 )( y  1 )
y 1
3( y
 3  y  1)
( 3 y  2 )( y  1 )
1


3y
2

1( y  1)
( y  1 )( y  1 )
 y  2
( y  1 )( y  1 )


(3 y  2 )
( y  1)
9
Subtracting Algebraic Fractions
with Unlike Denominators
3x  2
x
2
 2 x  1
x

3x  2
( x  1 )( x  1 )
x
2

 1
 x
( x  1 )( x  1 )
( 3 x  2 )( x  1 )
( x  1 )( x  1 )( x  1 )
3x
2
 x ( x  1)

 3x  2 x  2  x
( x  1 )( x  1 )
2
 x
( x  1 )( x  1 )( x  1 )
2 x
2
 6 x  2
( x  1 )( x  1 )( x  1 )




2(x
2
 3 x  1)
( x  1 )( x  1 )( x  1 )
10
Complex Fraction
• Is a fraction that has a fractional numerator
and/or a fractional denominator.
• Two ways to simplify complex fractions:
1) Multiply numerator and denominator of the
complex fraction by the LCD
2) Combine fractions in the numerator and
combine fractions in the denominator and then
divide the two fractions.
11
Simplify the complex fraction
Ex 1:
x
2
 5x  6
2x
x
2
2x
2
x
y
2
x
y
2
2x
( x  3 )( x  2 )
( x  3 )( x  3 )
Ex 2:
 5x  6
2x

 9
2

2
y
 9
2
x
x
2
2
 x  6
 5x  6
y

2x
2

y
x
2
 5x  6
x
2
 9

1
(x  2)
( x  3)
6 

 x 1

x 


2
1
y


x


6
6  

x  5 
 x  5 
 
x
x  

x 1
6
2x
x 

1 

x 

1 
( x  3 )( x  2 )
( x  3 )( x  2 )

x  2
x  2
12