Question

factorize : x^6 + 64x^3 - 64

Answer 1

Equation at the end of step  1  :
  (x6) -  26x3 Step  2  :
 Step  3  :
Pulling out like terms :
 3.1     Pull out like factors :
   x6 - 64x3  =   x3 • (x3 - 64) 
Trying to factor as a Difference of Cubes:
 3.2      Factoring:  x3 - 64 
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 :  64  is the cube of   4 
Check :  x3 is the cube of   x1

Factorization is :
             (x - 4)  •  (x2 + 4x + 16)