Simplifying Rational Expressions

When dealing with rational equations, it comes in handy to simplify the expressions in order to identify some of the characteristics of these rational functions, and to manipulate these expressions.

In order to simplify rational expressions, factor the numerator and denominator first. Then eliminate all of the multiplicative identity pairs to simplify.

Example 1: 60 126

2235 2337 Expand the numerator and denominator with prime factorization.

2235 2337 Simplify by eliminating all multiplicative identity pairs.

25 37 = 10 21 Rewrite in simplest terms.


Example 2: 7 x 5 y 3 21 x 2 y 4

7xxxxxyyy 37xxyyyy Expand the numerator and denominator.

7xxxxxyyy 37xxyyyy Simplify by eliminateing identity pairs.

x 3 3y         Rewrite in simplest terms.


Example 3: 4 x 2 24 x 2 8x

4xx 4x( 6x2 ) Expand the numerator and factor the denominator.

4xx 4x( 6x2 ) Simplify.

x 6x2          Rewrite.


Sometimes the rational expression is already in simplest form.

Example: 5 x( x+3 ) There are no common factors between 5 and x(x + 3), so it cannot be simplified.

Simplify by factoring binomials

Example 4: 30x+15 5x+20

15( x+3 ) 5( x+4 ) Factor the numerator and denominator by their respective GCFs.

35( x+3 ) 5( x+4 )      Simplify.

3( x+3 ) x+4     Rewrite.


Example 5: x 2 11x+24 x 2 3x40

( x8 )( x3 ) ( x8 )( x+5 ) Factor the numerator and denominator.

( x-8 )( x-3 ) ( x-8 )( x+5 ) Simplify.

x3 x+5      Rewrite.


Example 6: x 3 5 x 2 3x+15 x 2 8x+15

x 2 ( x5 )3( x5 ) ( x5 )( x3 ) Factor four terms by factoring two at a time

( x 2 3 )( x5 ) ( x5 )( x3 )     The other factor follows.

( x 2 -3 )( x-5 ) ( x-5 )( x-3 )    Simplify.

x 2 3 x3        Rewrite.


Now that you know how to simplify rational expressions, you will be able to perform addition and subtraction with rational expressions.



Related Links:
Math
algebra
Simplifying the Rational Expressions Worksheets
Rational Expressions
Adding Rational Expressions
Multiplying Rational Equations
Algebra Topics


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