Simplify Rational Expressions

      4x - 2
     --------
      2x - 1

Detailed Solution to Example 1

• Factor both the numerator and denominator completely.
  

       2(2x - 1)
       ---------
         2x - 1
  

  
  
  
• Cancel common factors to reduce and simplify the given expression.
  

        2(2x - 1)
      = -------- = 2 , with x not equal to 1
          2x - 1
  

  
  
  Note: The given expression is equal to 2 on the condition that x is not equal to 1. x = 1 is not in the domain of the expression.

Example 2: Simplify the rational expression

      4x + 16
     ---------
      x2 - 16

Detailed Solution to Example 2

• Factor both the numerator and denominator completely.
  

        4x + 16         4(x + 4)
       --------- = ----------------
        x2  - 16      (x + 4)(x - 4)
  

  
  
  
• Cancel common factors to simplify the expression.
  

           4(x + 4)
       = ---------------
          (x - 4)(x + 4)
  
  
  
  
           4
       = --------    , with x not equal to -4.
          (x - 4)
  

Example 3: Simplify the rational expression

       x2 + x - 2
     -------------
      -x2 -2x + 3

Detailed Solution to Example 3

• Factor both the numerator and denominator completely.
  

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

  
  
  
• Note that - x + 1 = - (x - 1) in the denominator.
  

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

  
  
  
• Cancel common factors.
  

          (x + 2)(x - 1)
       = ---------------
         -(x + 3)(x - 1)
  
  
  
  
            (x + 2)
       = - --------   , with x not equal to 1
            (x + 3)
  
  

Exercises

Simplify the rational expression.

1.            4x + 2
        ---------------
         (x - 9)(2x + 1)
   

   
   

2.        x2 + 7x + 12
        ------------------
         -x2 - 5x - 6
   
   

   
   

3.            x4 - 1 
        ----------------
         (x2 + 1)(x - 1)
   
   

   
   

4.        x3 + x2 - 2x 
         ---------------
             x - 1
   
   

Answers to the above exercises

• Definition and domain of Rational Expressions.