Algebraic Fractions
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Simplify Rational Expressions
Multiply Rational Algebraic
Expressions
Add Rational Algebraic Expressions
Solve Rational Equations
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
a
mn

a
n
a
a 1
0
a
m
1
 m
a
Simplify. Reduce the fractions.
4
14m
9
7m
27mmmm
1  7  m m m m m m m m m
2
 5
m
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.
Simplify. Reduce the fractions.
x  4x  4
2
4 x
2
 x  2  x  2 
2  x  2  x 

  x  2
x2
 
2 x
Multiply these rational algebraic
expressions.
Fraction Factor, Factor Fraction !
Multiply numerators to numerators, and
denominators to denominators.
Beware of addition – use parentheses
Beware of “backwards” subtraction
Multiply these rational algebraic
expressions.
12 x  3x 8 x  12

10 x  15 9 x  18
2
3x(4 x  1) 4(2 x  3)

5(2 x  3) 9( x  2)
3
4 x(4 x  1)

15( x  2)
Divide these rational algebraic
expressions.
Divide Fractions, no Don’t!!!
Multiply by the reciprocal of the fraction
behind the 
Beware of addition – use parentheses
Beware of “backwards” subtraction
Divide these rational algebraic
expressions.
49  x
5
x y
 ( x  7)
2
x  14 x  49

3 4
x y
2
y3
(7  x)(7  x)
x y
 y (7  x)

 2
5
x y
( x  7)( x  7)
x ( x  7)
x2
3
4
3
Find the LCM of the expressions.
Finding the LCM:
M: Find all the prime factors. ( plus sign use parentheses )
C: Then write down all the Common factors.
L: Then write down all the Leftover factors.
Find the LCM of the expressions.
M:
M:
4xy2
22xyy
22xyy
12xy2 + 6y
6 y (2xy + 1)
2 3 y(2xy + 1)
C:
2y
L:
2 x y 3 (2xy + 1)
LCM: 2 y [ 2 x y 3 (2xy + 1)] = 12xy2 (2xy + 1)
Add these algebraic fractions.
Always factor the DENOMINATOR first.
Find the LCM, use every factor, only as many times as necessary.
Make ONE common denominator, make the bottoms the “same”.
FIX the numerators, multiply both the numerator and
denominator by the missing factors of the LCM.
Multiply out everything in the numerator.
Then combine like terms. Beware of subtraction!!!
Factor the numerator to see if you can REDUCE the fraction.
Add these algebraic fractions.
x
53
 2
3x  6 x  2 x 
3  x  2 x  x  2 3
x
x
x
 15
3 x  x  2
2
Subtract these algebraic fractions.
Always factor the DENOMINATOR first.
Find the LCM, use every factor, only as many times as necessary.
Make ONE common denominator, make the bottoms the “same”.
FIX the numerators, multiply both the numerator and
denominator by the missing factors of the LCM.
Beware of subtraction!!! Change all the signs in the following numerator.
Multiply out everything in the numerator. Then combine like terms.
The last step is to factor the numerator to see if you can
REDUCE the fraction.
Subtract these algebraic fractions.
5x
2  x  2

2
x  2 x  8  x  4  x  2 
 x  4 x  2
5x 
  2x  4
 x  4 x  2
3x  4

 x  4 x  2
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.
Solve the Rational Equation.
x 5
2
Restrictions : x = -9

x 9
5
Multiply by the LCM: 5(x + 9) Reduce to clear fractions
Solve:
5x – 25 = 2x + 18
– 2x
-2x
.
3x – 25 =
18
+
25
+ 25
3x =
43
(1/3) 3x = (1/3) 43
x = 43
3
 x  5 x  9 5
x 9
2  x  9  5

5