Question

Determine all possible values of the expression a^3+b^3+c^3-3abc

where a,b and c are non negative integers​

Answer 1

Step-by-step explanation:

First value of the expression:

a³+b³+c³-3abc

= (a³+b³)+c³-3abc

= (a+b)³-3ab(a+b)+c³-3abc

= (a+b)³+c³-3ab(a+b+c)

= (a+b+c){(a+b)²-(a+b).c+c²}-3ab(a+b+c)

=(a+b+c){(a²+2ab+b²)-ca-bc+c²}-3ab(a+b+c)

= (a+b+c)(a²+b²+c²+2ab-ca-bc-3ab)

= (a+b+c)(a²+b²+c²-ab-bc-ca).

Second value:

a³+b³+c³-3abc

= (a+b+c)(a²+b²+c²-ab-bc-ca)

= 1/2 (a+b+c)(2a²+2b²+2c²-2ab-2bc-2ca)

= 1/2 (a+b+c){(a²-2ab+b²)+(b²-2bc+c²)+(c²-2ca+a²)}

= 1/2 (a+b+c){(a-b)²+(b-c)²+(c-a)²}.