Question

factorize
3acube - 243abcube

Answer 1

Step by step solution :Step  1  :Equation at the end of step  1  : (3 • (a3)) - 35ab3 Step  2  :Equation at the end of step  2  : 3a3 - 35ab3 Step  3  :Step  4  :Pulling out like terms :

 4.1     Pull out like factors :

   3a3 - 243ab3  =   3a • (a2 - 81b3) 

Trying to factor as a Difference of Cubes:

 4.2      Factoring:  a2 - 81b3 

Theory : A difference of two perfect cubes,  a3 - b3 can be factored into
              (a-b) • (a2 +ab +b2)

Proof :  (a-b)•(a2+ab+b2) =
            a3+a2b+ab2-ba2-b2a-b3 =
            a3+(a2b-ba2)+(ab2-b2a)-b3 =
            a3+0+0+b3 =
            a3+b3

Check :  81  is not a cube !! 
Ruling : Binomial can not be factored as the difference of two perfect cubes

Final result : 3a • (a2 - 81b3)