Multiplication and Division of Rational Expressions   Simplifying Rational Expressions We define a Rational Expression as a fraction where the numerator and the denominator are polynomials in one or more variables.   Examples    x2 - y2                           is a rational expression. (x - y)2   To simplify, we just factor and cancel: (x - y)(x + y)            x  + y                          =                     (x - y)2                  x - y   3x2 - 4x              x(3x - 4)             3x - 4                     =                        =                   2x2 - x                x(2x - 1)             2x - 1   Exercises:     2x2 - x - 1                                       3x2 - 2x - 1 8x2 + 18x + 9                                     12x2 + 17x + 6 x2 - xy - 2x + 2y                                         x2 - xy - 3x + 3y Multiplication of Fractions Recall that when we multiply fractions we first cross cancel.  For example:         8   21         4    7         28                     =                =               9   10         3    5         15    When we multiply rational expressions we follow the same approach.  First we factor then we cross cancel.    Examples:   32a2     21x2b4                                                                        Cross Cancel   7xb2          8a3 x2 - 2x + 1       x2 + 4x + 3                                                         First Factor      x  + 1               x - 1          (x - 1)2       (x + 3)(x + 1)  =                                                     Cancel the x + 1 and the x - 1            x + 1            x - 1 =   (x - 1)(x + 3) ExercisesMultiply and simplify   14x2          9y                                                           21x3y3       8x 2x2 - 5x - 12         3x2 -14x - 5                                                                3x2 - 11x - 4         2x2 - 7x - 15 9x2 - 6xy + y2        6x3y                                                               2xy2 -6x2y          3x - y Division of Fractions Recall that when we divide fractions we multiply by the reciprocal.   For example:            8                                           9                     8     21              4      7             28                            =                      =                        =                    10                    9     10              3       5            15                                21 For rational functions we do the same thing.  To divide rational functions, multiply by the reciprocal and then factor and cross cancel.   Example            17a2b3                                                      18yx                       17a2b3       9xy                                             =                                   =     b3                 34a2                        18yx        34a2                                                        9xy            x2 - x - 2                                                           x + 3                              x2 - x - 2       3x + 9                                                   =                                                        2x + 2                              x + 3             2x + 2                                                          3x + 9           (x - 2)(x + 1)           3(x + 3)             3x - 6        =                                                        =                                     x + 3                 2(x + 1)                  2           Exercises Divide.            17a2b3                                                    18xy                                                                            34a2                                                    9xy 32x2c                                                               12a2b                                                                       17a2x2       16x                                               34b2c         9a 8x + 4                                                                            x - 3                                                                2x2 - 5x - 3 x2 - 3x - 18                                                                x2 + 5x + 6                                                                        x2 - 8x + 12                                                                   x2  -  4 