Solution - Factoring binomials using the difference of squares

Step  1  :

Trying to factor as a Difference of Squares :

 1.1      Factoring:  x⁶⁵⁸⁸-r⁶ 

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 :  x⁶⁵⁸⁸  is the square of  x³²⁹⁴ 

Check :  r⁶  is the square of  r³ 

Factorization is :       (x³²⁹⁴ + r³)  •  (x³²⁹⁴ - r³) 

Trying to factor as a Sum of Cubes :

 1.2      Factoring:  x³²⁹⁴ + r³ 

Theory : A sum 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 :  x³²⁹⁴ is the cube of   x¹⁰⁹⁸

Check :  r³ is the cube of   r¹

Factorization is :
             (x¹⁰⁹⁸ + r)  •  (x²¹⁹⁶ - x¹⁰⁹⁸r + r²) 

Trying to factor as a Sum of Cubes :

 1.3      Factoring:  x¹⁰⁹⁸ + r 

Check :  x¹⁰⁹⁸ is the cube of   x³⁶⁶

Check :  r ¹ is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Trying to factor a multi variable polynomial :

 1.4    Factoring   x²¹⁹⁶ - x¹⁰⁹⁸r + r²

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

 Factorization fails

Trying to factor as a Difference of Squares :

 1.5      Factoring: x³²⁹⁴ - r³

Check :  x³²⁹⁴  is the square of  x¹⁶⁴⁷ 

Check :  r³   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:

 1.6      Factoring: x³²⁹⁴ - r³

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 :  x³²⁹⁴ is the cube of   x¹⁰⁹⁸

Check :  r³ is the cube of   r¹

Factorization is :
             (x¹⁰⁹⁸ - r)  •  (x²¹⁹⁶ + x¹⁰⁹⁸r + r²) 

Trying to factor as a Difference of Squares :

 1.7      Factoring:  x¹⁰⁹⁸ - r 

Check :  x¹⁰⁹⁸  is the square of  x⁵⁴⁹ 

Check :  r¹   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:

 1.8      Factoring:  x¹⁰⁹⁸ - r 

Check :  x¹⁰⁹⁸ is the cube of   x³⁶⁶

Check :  r ¹ is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Trying to factor a multi variable polynomial :

 1.9    Factoring   x²¹⁹⁶ + x¹⁰⁹⁸r + r²

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

 Factorization fails

Final result :

  (r+x1098)•(x2196+r2-x1098r)•(x1098-r)•(x2196+r2+x1098r)