Adding Rational Expressions

Adding and subtracting rational expressions are similar to adding and subtracting numerical ratios.

In order to add or subtract a rational expression, a common denominator must be found first, and then the operation can be carried out in the numerator.

With like denominators, simply add the two numerators to find the sum.

Example 1: 4x+1+xx+1  

4x+1+xx+1    =    x+4x+1   Add the numerators (the denominator does not change).


With unlike denominators

First find the common denominator

By multiplying denominators

Example 2: Single Term Ratios
32x+57

77×32x+2x2x×57     Multiply each ratio by one using the other denominator.

2114x+10x14x Multiply across.

10x+2114x      Add the numerators.


By finding least common multiple of the denominators

Example 3: Single Term Ratios
78x+165x

The least common multiple (LCM) from 8x and 5x is 40x

55×78x  +  88×165xMultiply the ratios by one to get a common denominator.

3540x+12840x Multiply across.

16340x   Add the numerators, simplify if possible.

163 and 40 are relatively prime, so this ratio cannot be simplified.


Example 4: Factoring trinomials in the denominator.

2xx2 +5x24 + 4 x3

x2 + 5x - 24 = (x + 8)(x - 3)Factor the denominator.

       The LCM is (x + 8)(x - 3).

2x ( x+8 )( x3 ) + ( x+8 ) ( x+8 ) × 4 ( x3 ) Multiply the second ratio to obtain a common denominator.

2x+4( x+8 ) ( x+8 )( x3 ) Add the numerators.

2x+4x+32 ( x+8 )( x3 ) Use the distributive property to combine like terms.

6x+32 ( x+8 )( x3 ) Rewrite in simplified form.


Example 5: Special products in the denominator.

x x 2 4 + x+2 x 2 +4x+4

x2 - 4 = (x + 2)(x - 2) and x2 + 4x + 4 = (x + 2)(x + 2)Factor the denominators.

       The LCM is (x + 2)(x + 2)(x - 2).

( x+2 ) ( x+2 ) × x ( x+2 )( x2 )   +   ( x2 ) ( x2 ) × ( x+2 ) ( x+2 )( x+2 )   Multiply the ratios to obtain common denominators.

x( x+2 ) ( x+2 )( x+2 )( x2 ) + ( x2 )( x+2 ) ( x2 )( x+2 )( x+2 ) Rewrite.

x( x+2 )+( x2 )( x+2 ) ( x+2 )( x+2 )( x2 ) Add the numerators.

( x+( x2 ) )( x+2 ) ( x+2 )( x+2 )( x2 )    Regroup the terms to factor the numerator.

( 2x2 )( x+2 ) ( x+2 )( x+2 )( x2 )   Eliminate common factors.

( 2x2 ) ( x+2 )( x2 )     Rewrite in simplified form.


Example 6: Subtraction
x+3 x 2 2x8 x5 x 2 12x+32

x2 - 2x - 8 = (x - 4)(x + 2)Factor the denominators.

          The LCM is (x - 8)(x - 4)(x + 2).

( x8 ) ( x8 ) × ( x+3 ) ( x4 )( x+2 ) ( x+2 ) ( x+2 ) × ( x5 ) ( x8 )( x4 )    Multiply the ratios to obtain common denominators.

x 2 5x24 ( x8 )( x4 )( x+2 ) x 2 3x10 ( x8 )( x4 )( x+2 ) Multiply to determine new numerators

2x14 ( x8 )( x4 )( x+2 )         Subtract the second numerator from the first.

With -2x - 14 = -2(x + 7)    there are no common factors to simplify the ratio; keep the ratio as is.


Example 7: Addition and subtraction together

x+3 x 2 25 + x1 x5 2 x+5

x2 - 25 = (x + 5)(x - 5)Factor the denominator.

The LCM is (x + 5)(x - 5).

( x+3 ) ( x5 )( x+5 ) + ( x+5 ) ( x+5 ) × ( x1 ) ( x5 ) ( x5 ) ( x5 ) × 2 ( x+5 )      Multiply the ratios to obtain a common denominator.

  ( x+3 )+( x 2 +4x5 )( 2x10 ) ( x+3 )( x5 )( x+5 )     Rewrite the numerator.

x 2 +3x+8 ( x+3 )( x5 )( x+5 )           Combine like terms.


Now that you can add and subtract rational expressions, you are ready to start solving rational equations.



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


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