Question

What is the factored form of 8x^24-27y^6

Answer 1

 8x24-27y6 Final result : (2x8 - 3y2) • (4x16 + 6x8y2 + 9y4)

Step by step solution : Step  1  :Skip Ad
Equation at the end of step  1  : (8 • (x24)) - 33y6 Step  2  :Equation at the end of step  2  : 23x24 - 33y6 Step  3  :Trying to factor as a Difference of Squares :

 3.1      Factoring:  8x24-27y6 

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A2 - AB + BA - B2 =
         A2 - AB + AB - B2 =
         A2 - B2

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  8  is not a square !!

Ruling : Binomial can not be factored as the
difference of two perfect squares

Trying to factor as a Difference of Cubes:

 3.2      Factoring:  8x24-27y6 

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 :  8  is the cube of  2 

Check :  27  is the cube of   3 
Check :  x24 is the cube of   x8

Check :  y6 is the cube of   y2

Factorization is :
             (2x8 - 3y2)  •  (4x16 + 6x8y2 + 9y4) 

Trying to factor as a Difference of Squares :

 3.3      Factoring:  2x8 - 3y2 

Check :  2  is not a square !!

Ruling : Binomial can not be factored as the
difference of two perfect squares

Trying to factor a multi variable polynomial :

 3.4    Factoring    4x16 + 6x8y2 + 9y4 

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

 Factorization fails

Final result : (2x8 - 3y2) • (4x16 + 6x8y2 + 9y4)

Answer 2

The answer will be (2x8 - 3y2) • (4x16 + 6x8y2 + 9y4)