Solution - Reducing fractions to their lowest terms

Step  1  :

            y2
 Simplify   ——
            x 

Equation at the end of step  1  :

                          y2      
  (((x2) +  5xy) -  (24 • ——)) -  3y
                          x       

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

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

                 x2 + 5xy     (x2 + 5xy) • x
     x2 + 5xy =  ————————  =  ——————————————
                    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

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   x² + 5xy  =   x • (x + 5y) 

Adding fractions that have a common denominator :

 3.2       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 • (x+5y) • x - (24y2)     x3 + 5x2y - 24y2 
 ———————————————————————  =  ————————————————
            x                       x        

Equation at the end of step  3  :

  (x3 + 5x2y - 24y2)     
  —————————————————— -  3y
          x             

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

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

          3y     3y • x
    3y =  ——  =  ——————
          1        x   

Trying to factor a multi variable polynomial :

 4.2    Factoring    x³ + 5x²y - 24y² 

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

 Factorization fails

Adding fractions that have a common denominator :

 4.3       Adding up the two equivalent fractions

 (x3+5x2y-24y2) - (3y • x)      x3 + 5x2y - 3xy - 24y2 
 —————————————————————————  =  ——————————————————————
             x                           x           

Checking for a perfect cube :

 4.4    x³ + 5x²y + 3xy + 24y²  is not a perfect cube

Final result :

 x3 + 5x2y + 3xy + 24y2