Solution - Other Factorizations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x3"   was replaced by   "x^3". 

Step  1  :

Equation at the end of step  1  :

  (2 • (x2)) -  24x32
 Step  2  :

Equation at the end of step  2  :

  2x2 -  24x32

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   2x² - 16x³²  =   -2x² • (8x³⁰ - 1) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  8x³⁰ - 1 

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:

 4.3      Factoring:  8x³⁰ - 1 

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 :  1  is the cube of   1 
Check :  x³⁰ is the cube of   x¹⁰

Factorization is :
             (2x¹⁰ - 1)  •  (4x²⁰ + 2x¹⁰ + 1) 

Trying to factor as a Difference of Squares :

 4.4      Factoring:  2x¹⁰ - 1 

Check :  2  is not a square !!

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

Trying to factor by splitting the middle term

 4.5     Factoring  4x²⁰ + 2x¹⁰ + 1 

The first term is,  4x²⁰  its coefficient is  4 .
The middle term is,  +2x¹⁰  its coefficient is  2 .
The last term, "the constant", is  +1 

Step-1 : Multiply the coefficient of the first term by the constant   4 • 1 = 4 

Step-2 : Find two factors of  4  whose sum equals the coefficient of the middle term, which is   2 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result :

  -2x2 • (2x10 - 1) • (4x20 + 2x10 + 1)