Multiply & Divide Rational Expressions
Reducing Fractions:
Always look at the FACTORS.
If there is addition, ALWAYS factor first!!!
Then reduce any factor in the numerator with the same factor in the denominator.
10 x 7 y 3
6x3 y
x2  4
x2  4x  4
If subtraction is written backwards, factor out a negative.
When you factor out the negative sign, you write the subtraction switched around.
x2 1
1 x
x2  2 x  3
9  x2
Multiply Fractions: Multiply numerators to numerators, denominators to denominators.
You may reduce first, any factor in the numerator with the same factor in the
denominator.
8 x  12 42 x  21

14 x  7 32 x  48
x2  4
9 x  18

x 2  4 x  4 6  3x
Divide Fractions:
Don’t!!! Multiply by the reciprocal of the fraction behind the
x2 y5
xy 6

x 2  11x  30 x 2  7 x  10

x2  5x  6 x2  2 x  1

x3
x2  9
Adding Algebraic Fractions
Adding Algebraic Fractions:
1
5

3x  6 x  2
Always factor the DENOMINATOR first.
1
5

3( x  2)
( x  2)
Use every factor,
only as many times as necessary.
Make ONE common denominator,
make the bottoms the “same”.
3( x  2)
FIX the numerators.
Multiply out everything in the numerator. Then combine like terms. Beware of subtraction!!!
The last step is to factor the numerator to see if you can REDUCE the fraction.
5
3

3a 4a
5
x
 2
x2 x 4
Subtract, change to addition AND change all the signs in the numerator.
x  5 3x 1

3x2
6x
3x
2

x4 x6
Solve Rational Equations
An EQUATION must contain an equal sign.
ALGEBRAIC rational expression contains one or more variables in the denominator.
When solving a rational equation, you must remove the variable from the denominator.
Just as before, there are 3 types of solutions:
A unique solution
An infinite number of solutions
No solution
Recall that division by zero is undefined, therefore the variable can NOT take on values that cause the
denominator to be zero. We restrict the values that the variable can have by determining what values for
the variable would cause the denominator to be zero. Always identify the restrictions for the variable
before you solve the problem.
1. Clear the fraction, Multiply EACH TERM by the
LCM.
2. Reduce each term, so that no fractions exist.
3. Get just 1 variable.
Combine like terms and/or get all the variables on 1 side of the equal sign.
4. Isolate the variable.
Add the opposite and/or multiply by the reciprocal.
4
7
1 
x2
x2
Solve.
y
y4
4y 3

 2
y 3 y  2 y  y 6
Solve.
50
30

r 2
r