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

 

  1.  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


  2.   3x2 - 4x              x(3x - 4)             3x - 4
                        =                        =                
      2x2 - x                x(2x - 1)             2x - 1

 

Exercises:

 

  1.   2x2 - x - 1
                                       
      3x2 - 2x - 1

    (2x + 1) / (3x + 1)

  2.   8x2 + 18x + 9
                                      
     12x2 + 17x + 6

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

  3.  x2 - xy - 2x + 2y
                                          
     x2 - xy - 3x + 3y

    (x - 2) / (x - 3)


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:

 

  1. 32a2     21x2b4           
                              
                                      Cross Cancel  
    7xb2          8a3                   




  2.  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)


Exercises

Multiply and simplify

 

  1. 14x2          9y
                                                             
    21x3y3       8x

    3 / (4x^2 y^2)

  2.  2x2 - 5x - 12         3x2 -14x - 5
                                                    
                  
    3x2 - 11x - 4         2x2 - 7x - 15

    1

  3.  9x2 - 6xy + y2        6x3y
                                                             
        2xy2 -6x2y          3x - y

-3x^2


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


  1.            17a2b3      
                                     
                 18yx                       17a2b3       9xy         
                                       =                                   =     b3  
                  34a2                        18yx        34a2           
                                 
                  9xy

  2.            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.

  1.            17a2b3      
                                    
                18xy         
                                                        
                 34a2           
                              
                 9xy

    b^3/4

  2.                  32x2c      
                                          
                     12a2b         
                                                       
             17a2x2       16x
                                       
              34b2c         9a 

    (3bc^2) / (a^3 x)

  3.                  8x + 4      
                                                      
                      x - 3           
                                          
                 2x2 - 5x - 3

    4(2x+1)^2

  4.              x2 - 3x - 18      
                                               
                 x2 + 5x + 6         
                                                    
                 x2 - 8x + 12           
                                          
                    x -  4

    1


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