This document provides information and examples about algebraic fractions, including: - Simplifying and reducing rational expressions by dividing both the numerator and denominator by common factors. - Multiplying, dividing, adding, and subtracting rational algebraic expressions by using common denominators. - Finding the least common multiple of denominators. - Solving rational equations by clearing fractions, combining like terms, and isolating the variable.

1. Algebraic FractionsAlgebraic Fractions
• Simplify Rational Expressions
• Multiply Rational Algebraic Expressions
• Add Rational Algebraic Expressions
• Solve Rational Equations

2. Simplify. Reduce the fractions.Simplify. Reduce the fractions.
Always look at the FACTORS.
When reducing fractions,
divide both the numerator and the denominator by the same
factor.
Exponent Rules:
m
m n
n
a
a
a
−
= 0
1a =
1m
m
a
a
−
=

3. 1 7
2 7
mmmmmmmmm
mmmm
− • •
• •
Simplify. Reduce the fractions.Simplify. Reduce the fractions.
4
9
14
7
m
m−
2 7
1 7
mmmm
mmmmmmmmm
• •
− • • 5
2
m
−
=

4. Simplify. Reduce the fractions.Simplify. Reduce the fractions.
Reducing Fractions: Always look at the FACTORS.
If there is addition, ALWAYS factor first,
use parentheses!!!
Then reduce any factor in the numerator with the same
factor in the denominator.
If subtraction is written backwards, factor out a negative.
When you factor out the negative sign,
you write the subtraction switched around.

5. ( ) ( )
( ) ( )2
2 2
2
x x
x x
− −
− +
Simplify. Reduce the fractions.Simplify. Reduce the fractions.
2
2
4 4
4
x x
x
− +
−
( ) ( )
( ) ( )
2 2
2 2
x x
x x
− −
− +
x 2
2 x
−
=
+
−
( )2− −x

6. Multiply these rational algebraicMultiply these rational algebraic
expressions.expressions.
Fraction Factor, Factor Fraction !Fraction Factor, Factor Fraction !
Multiply numerators to numerators, and denominators toMultiply numerators to numerators, and denominators to
denominators.denominators.
Beware of addition – use parenthesesBeware of addition – use parentheses
Beware ofBeware of ““backwardsbackwards”” subtractionsubtraction

7. Multiply these rational algebraicMultiply these rational algebraic
expressions.expressions.
2
12 3 8 12
10 15 9 18
x x x
x x
+ −
•
− +
3 (4 1) 4(2 3)
5(2 3) 9( 2)
x x x
x x
+ −
•
− +
3
4 (4 1)
15( 2)
x x
x
+
=
+
3 (4 1)
5(2 3)
x x
x
+
−

8. Divide these rational algebraicDivide these rational algebraic
expressions.expressions.
Divide Fractions, noDivide Fractions, no DonDon’’t!!!t!!!
Multiply by the reciprocal of the fraction behind theMultiply by the reciprocal of the fraction behind the
Beware of addition – use parenthesesBeware of addition – use parentheses
Beware ofBeware of ““backwardsbackwards”” subtractionsubtraction
÷

9. Divide these rational algebraicDivide these rational algebraic
expressions.expressions.
2 2
5 3 4
49 14 49x x x
x y x y
− − +
÷
3 4
5
(7 )(7 )
( 7)( 7)
x x x y
x y x x
− +
•
− −
( 7)x− −
x2
y3
3
2
(7 )
( 7)
y x
x x
− +
=
−

10. Find the LCM of the expressions.Find the LCM of the expressions.
Finding theFinding the LLCCM:M:
M: Find all the prime factors. (M: Find all the prime factors. ( plus sign use parenthesesplus sign use parentheses ))
CC: Then write down all the: Then write down all the CCommon factors.ommon factors.
LL: Then write down all the: Then write down all the LLeftover factors.eftover factors.

11. Find the LCM of the expressions.Find the LCM of the expressions.
4xy4xy22
12xy12xy22
+ 6y+ 6y
M: 2 2 x y y 6 y (2xy + 1)M: 2 2 x y y 6 y (2xy + 1)
M:M: 22 2 x y2 x y yy 22 33 yy(2xy + 1)(2xy + 1)
CC:: 2 y2 y
LL: 2 x y 3 (2xy + 1): 2 x y 3 (2xy + 1)
LLCCM:M: 2 y2 y [ 2 x y 3 (2xy + 1)][ 2 x y 3 (2xy + 1)]
= 12xy= 12xy22
(2xy + 1)(2xy + 1)

12. Add these algebraic fractions.Add these algebraic fractions.
Always factor the DENOMINATOR first.Always factor the DENOMINATOR first.
Find the LCM,Find the LCM, use every factor, only as many times as necessary.use every factor, only as many times as necessary.
MakeMake ONEONE common denominator,common denominator, make the bottoms themake the bottoms the ““samesame””..
FIXFIX the numerators, multiply both the numerator andthe numerators, multiply both the numerator and
denominator by the missing factors of the LCM.denominator by the missing factors of the LCM.
Multiply out everything in the numerator.Multiply out everything in the numerator.
Then combine like terms.Then combine like terms. Beware of subtractionBeware of subtraction!!!!!!
Factor the numerator to see if you can REDUCE the fraction.Factor the numerator to see if you can REDUCE the fraction.

13. Add these algebraic fractions.Add these algebraic fractions.
2
5
3 6 2
x
x x x
+
+ +
( )2x x +( )3 2x +
( )23x x
+
+
x
x
( )
2
3
15
2
x
x x
+
+
3
3•
•

14. Subtract these algebraic fractions.Subtract these algebraic fractions.
Always factor the DENOMINATOR first.Always factor the DENOMINATOR first.
Find the LCM,Find the LCM, use every factor, only as many times as necessary.use every factor, only as many times as necessary.
MakeMake ONEONE common denominator,common denominator, make the bottoms themake the bottoms the ““samesame””..
FIXFIX the numerators, multiply both the numerator andthe numerators, multiply both the numerator and
denominator by the missing factors of the LCM.denominator by the missing factors of the LCM.
Beware of subtractionBeware of subtraction!!!!!! Change all the signs in the following numerator.Change all the signs in the following numerator.
Multiply out everything in the numerator.Multiply out everything in the numerator. Then combine like terms.Then combine like terms.
The last step is to factor the numerator to see if you canThe last step is to factor the numerator to see if you can
REDUCE the fraction.REDUCE the fraction.

15. Subtract these algebraic fractions.Subtract these algebraic fractions.
2
5 2
2 8 4
x
x x x
−
+ − +
( ) ( )4 2x x
−
+ −
( )4x +
( ) ( )4 2x x+ −
( )2x −
( )2x −
( ) ( )4 2x x
+
+ −( ) ( )
5 2 4
4 2
x x
x x
+ − +
+ −
( ) ( )
3 4
4 2
x
x x
+
=
+ −

16. Solve each of these
Rational Equations.
ALGEBRAIC rational expression contains one or more
variables in the denominator. When solving a rational
equation, you must remove the variable from the
denominator.
Clear the fraction, Multiply EACH TERM by the LCM.
Reduce each term, so that no fractions exist.
Get just 1 variable.
Combine like terms and/or get all the variables on 1 side of the
equal sign.
Isolate the variable.
Recall that division by zero is undefined, therefore the variable
can NOT take on values that cause the denominator to be
zero. Be sure to identify the restrictions first.

17. Solve the Rational Equation.
Solve: Restrictions : x = -9
Multiply by the LCM: 5(x + 9) Reduce to clear fractions
5x – 25 = 2x + 18
– 2x -2x .
3x – 25 = 18
+ 25 + 25
3x = 43
(1/3) 3x = (1/3) 43
x = 43
3
5 2
9 5
−
=
+
x
x
( ) ( ) ( ) ( ) ( )95
9 5
525 9−
=
+
+ +x x
x
x