Rational Expressions



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The definition of rational expressions and their domains are examined.

Definition

A rational expression is the quotient of two polynomials.

Examples of Rational Expressions



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


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


  3.          1
         --------
          x2 + 1
    


  4.          x2 + 4
         --------------
          x3 + 2x2 - 3x 
    

Domain of Rational Expressions

It is clear that the above expressions are undefined if a division by 0 occurs. The domain of a rational expression excludes all values that make the denominator equal to 0.
  1. The domain of the rational expression
          x - 1 
         ------- 
          x + 2
    
    is the set of all real numbers except x = -2

  2. The domain of the rational expression
               x + 1
         ---------------
          (x + 2)(x - 3)
    
    is the set of all real numbers except x = -2 and x = 3

  3. The domain of the rational expression
             1
         --------
          x2 + 1
    
    is the set of all real numbers since the denominator x2 + 1 cannot be 0 for any real number x.

  4. To find the domain of the rational expression,
             x2 + 4
         --------------
          x3 + 2x2 - 3x 
    
    we first have to factor the denominator and find its zeros.
          x3 + 2x2 - 3x = x(x2 + 2x - 3)
    

    = x(x - 1)(x + 3)
    The domain of the above rational expressions is the set of all real numbers except x = 0, x = 1 and x = - 3.

Exercises

Find the domain of each of the Rational Expressions given below.

  1.       x + 9 
         ------- 
          x - 10
    


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


  3.          1
         --------
          x3 + 1
    


  4.       -x + 7
         --------
          x4 - 16
    

    Answers to the above exercises



More references to topics related to rational expressions.


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Updated: 26 November 2007 (A Dendane)