Is it possible to factor the polynomial Factor the polynomia

Is it possible to factor the polynomial?

Factor the polynomial. 64a^3 - 27b^6

Solution

given

64a^3 - 27b^6

Step  1  :

Equation at the end of step  1  :

Step 2 :

Equation at the end of step  2  :

Step  3  :

Trying to factor as a Difference of Squares :

3.1      Factoring:  64a³-27b⁶

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 :  64  is the square of  8
Check : 27 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:  64a³-27b⁶

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 :  27  is the cube of   3
Check : a³ is the cube of   a¹

Check : b⁶ is the cube of   b²

Factorization is :
             (4a - 3b²)  •  (16a² + 12ab² + 9b⁴)

Trying to factor as a Difference of Squares :

3.3      Factoring:  4a - 3b²

Check :  4  is the square of  2
Check : 3 is not a square !!

Factoring    16a² + 12ab² + 9b⁴ not possible

final result: