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
   
   
   

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

3.  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:

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
   
   
   

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

3.  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

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

   
   
   

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

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

   
   

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

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