Solution - Factoring multivariable polynomials

Step  1  :

            x2 - 2xy + y2
 Simplify   —————————————
                  x      

Trying to factor a multi variable polynomial :

 1.1    Factoring    x² - 2xy + y² 

Try to factor this multi-variable trinomial using trial and error 

 Found a factorization  :  (x - y)•(x - y)

Detecting a perfect square :

 1.2    x²  -2xy  +y²  is a perfect square 

 It factors into  (x-y)•(x-y)
which is another way of writing  (x-y)²

How to recognize a perfect square trinomial:  

 • It has three terms  

 • Two of its terms are perfect squares themselves  

 • The remaining term is twice the product of the square roots of the other two terms

Equation at the end of step  1  :

  (x - y)2    
  ———————— -  y
     x        

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  x  as the denominator :

         y     y • x
    y =  —  =  —————
         1       x  

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 (x-y)2 - (y • x)     x2 - 3xy + y2
 ————————————————  =  —————————————
        x                   x      

Trying to factor a multi variable polynomial :

 2.3    Factoring    x² - 3xy + y² 

Try to factor this multi-variable trinomial using trial and error 

 Factorization fails

Final result :

  x2 + 3xy + y2
  —————————————
        x