Rational expressions
    Remember, a rational number is any number that can be 
                           a
    written in the form    , where a and b are integers and b ≠ 0.
                           b
    Numbers written in this form are often called fractions.
    In algebra, a rational expression is an algebraic fraction 
                                         f(x)
    that can be written in the form        , where f(x) and g(x) are 
    polynomials and g(x) ≠ 0.            g(x)
    For example,
                      3             3x+1                 x3  2
                     x2             x2  2            x2 +3x 4
    For which values of x are each of the above expressions 
    undefined?
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  Rational expressions
    An algebraic fraction is undefined when the denominator is 0. 
    So,
          3
                  is undefined when x + 2 =     That is, when x = –2.
        x+2
        0.
        3x+1
                   is undefined when x2 – 2 = 0. That is, when x = ±√2.
        x2  2
          x3  2
                         is undefined when x2 + 3x – 4 = 0.
       x2 +3x 4
       We can factorize this to give (x + 4)(x – 1) = 0.
              x3  2
       So                    is undefined when x = –4 or x = 1.
            x2 +3x 4
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  Simplifying fractions by cancelling
    When the numerator and the denominator of a numerical 
    fraction contain a common factor, the fraction can be simplified 
    by cancelling. 
    For example, consider the fraction
                                282    2
                                    = 
                                423    3
    The highest common factor of 28 and 42 is _1_4_. 
    This fraction can therefore be written in its simplest terms by 
    dividing both the numerator and the denominator by 14. 
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   Simplifying algebraic fractions by cancelling
     Algebraic fractions can be cancelled in the same way.
     For example,
                                     2        3
                                 6a = 6aa                  3
                                     3 =                 ==
                                 8a       8aaa 4a
                                           4
     When the numerator or the denominator contains more than 
     one term, we have to factorize before cancelling. 
     For example,
                      3pq
      Simplify 15p 9p2
                      3pq        =      3pq        =      q
                  15p 9p2          3p(5 3p)          5 3p
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   Simplifying algebraic fractions by cancelling
                   3b 6
     Simplify b2  2b
                  3b 6 = 3(b 2) = 3
                  b2  2b      b(b 2)       b
                  x2 + x  2
      Simplify        2
                     x  1
                  x2 + x  2     (x 1)(x 2)        x2
                      2        =                   =
                    x  1         (x 1)(x 1)        x1
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