Three variables

----------
$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-bc-ca-ab)$
----------
$a^2(b-c)+b^2(c-a)+c^2(a-b)=-(a-b)(b-c)(c-a)$
----------
$a^3(b-c)+b^3(c-a)+c^3(a-b)=-(a-b)(b-c)(c-a)(a+b+c)$
----------
$(a+b+c)(a^7+b^7+c^7)-(a^3+b^3+c^3)(a^5+b^5+c^5)=(b+c)(a+b)(c+a)\left[2(a^5+b^5+c^5)-(a+b+c)(a^4+b^4+c^4)+\frac{1}{2}abc\{5(a^2+b^2+c^2)-(a+b+c)^2\}\right]$
----------
$10(a^2+b^2+c^2)\{(a+b+c)^3 - (a-b+c)^3 - (a+b-c)^3 + (a-b-c)^3\} = 3\{(a+b+c)^5 - (a-b+c)^5 - (a+b-c)^5 + (a-b-c)^5\}$