Addition and Subtraction of Rational Expressions

 

Review of Addition and Subtraction of Fractions

 

Example 1


          3           5               8               (2)(4)            2
                +              =              =                   =         
         20         20             20              (4)(5)            5

 

Notice the steps we have done to solve this problem.  We first combined the numerators since the denominators are the same.  Then we factored both the numerator and denominator and finally we cross cancelled.

 

Example 2

          1           2               3             4              7
                +              =             +             =         
          2           3               6             6              6

Notice that in this example, we needed to find a common denominator.

Addition and Subtraction of Rational Functions

To add and subtract rational functions, we follow the same method as fractions.  

 

  • Step 1  Factor everything and find the least common denominator.

  • Step 2  Multiply the numerators and the denominators by the appropriate denominator so that the denominator becomes the least common Denominator.

  • Step 3  Add the numerators together.

  • Step 4  Factor the numerator.

  • Step 5   Cancel any common factors.

 

Note:  Some of the steps will not be necessary.

 

Example

Perform the indicated operation.

          3x + 1             x      
                       +                  
          x2 - 1           x + 1        

 

Solution

  1.   3x + 1             x                   3x + 1                  x
                   +                 =                           +                
      x2 - 1           x + 1            (x - 1)(x + 1)          x + 1

    the LCD is 

            (x - 1)(x + 1)

  2.            3x + 1                 x         x - 1
     =                         +                          
          (x - 1)(x + 1)         x + 1      x - 1

               3x + 1                  x2  -  x
     =                         +                         
          (x - 1)(x + 1)         (x - 1)(x + 1)

  3.            x2 + 2x + 1       
     =                              
             (x - 1)(x + 1)   

  4.            (x + 1)2       
     =                          
           (x - 1)(x + 1)   

  5.          x + 1     
     =                
             x - 1 

Exercises

Perform the indicated operation.

  1.           3x             4x2      
                        -                                
              2y2            9y        

     (27x - 8x^2y) / (18y^2)

  2.           5              y      
                     +                                  
              y            y - 3        

    (y^2 + 5y -15) / (y(y - 3))

  3.           3x + 1          2x + 3      
                            +                           
               x - 2              2 - x        

    1

  4.           5x + 1              2x + 6      
                            +                           
               x + 2            x2 + 5x + 6        

     (5x + 3) / (x + 2)

  5.              2a - 9                  a + 27                      a + 7              
                                  +                          +                          
              a2 + a - 6            a2 - 2a - 15           a2 - 7a + 10        

    4(x + 1) / (x - 5)(x - 2)

  6.           3x + 1              2x + 1     
                            +                           
              x3 + 8           x2 - 2x + 4        

    (2x^2 + 7x +1) / (x + 2)(x^2 - 2x + 4)

 

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