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Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     x^3*y^3-(r^3)=0 

Step  1  :

Trying to factor as a Difference of Cubes:

 1.1      Factoring:  x³y³-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 :  x³ is the cube of   x¹

Check :  y³ is the cube of   y¹

Check :  r³ is the cube of   r¹

Factorization is :
             (xy - r)  •  (x²y² + xyr + r²) 

Trying to factor a multi variable polynomial :

 1.2    Factoring    x²y² + xyr + r² 

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

 Factorization fails

Equation at the end of step  1  :

  (xy - r) • (x2y2 + xyr + r2)  = 0 

Step  2  :

Theory - Roots of a product :

 2.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

 2.2     Solve   xy-r  = 0

In this type of equations, having more than one variable (unknown), you have to specify for which variable you want the equation solved.

We shall not handle this type of equations at this time.

Solving a Single Variable Equation :

 2.3     Solve   x²y²+xyr+r²  = 0

In this type of equations, having more than one variable (unknown), you have to specify for which variable you want the equation solved.

We shall not handle this type of equations at this time.

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