Which polynomial identity will prove that 19 = 27 − 8? difference of squares difference of cubes sum of cubes square of a binomial

The correct answer is:  [B]:  "Difference of Cubes". 


Explanation:

Note that the equation/identity for the "difference of cubes" is expressed as:


        " a³ − b³ = (a − b)(a² + ab + b²) "  ; 


Note the given equation:  " 19 = 27 − 8 " ;   →  (which is true). 


       The "right hand side" of this equation:

            →  " 27 − 8 " ;  contains two numbers:  
         
            →  "27" and "8" ;  both of which are "cubes" ; 

            → that is:  ∛27 = 3 ;   ↔  3³ = 3 * 3 * 3 = 9 * 3 = 27 ;  and:

                               ∛ 8  = 2 ;  ↔  2³ = 2 * 2 * 2 = 4 * 2 = 8 ; 

            → AND:  "8" is being SUBTRACTED from "27" ; 

            →  (hence, the "difference of squares" polynomial identity); 

So:  given:     " 19 = 27  − 8 " ; 

→  Rewrite as:

        " 19 =  3³  − 2³ " ; 

Now, consider the identity equation for the "difference of squares":

                          →   " a³ − b³ = (a − b)(a² + ab + b²) "  ; 


Take:  " 19  =  3³  − 2³ " ; 

and rewrite as:  

→  3³  −  2³  = 19 ; 

So:   (a³ − b³) = 3³  −  2³ ; 

a = 3 ;  b = 2 ; 

Plug in these values:

 " a³ − b³ = (a − b)(a² + ab + b²) " ; 

→   3³ − 2³  ≟  [3 − 2) [ 3² + (3*2) + 2² ]  ≟  19 ?  ; 

→  27 − 8   ≟  (1) (9 + 6 + 4) ≟  19 ?  ; 

→     19      ≟     (1) (15 + 4)   ≟  19 ?  ; 

→       19    ≟         (1) (19)     ≟  19 ?  ; 

→         19  ≟                19      ≟    19 ? ; 

→          19  =   19   =  19 !  Yes!