Question

$a + b + c$
)(
${a }^{2} + {b}^{2} + {c}^{2} - ab - ac - bc$
​

Answer 1

Remember this one. It's used vastly.

$(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) = a^3 + b^3 + c^3 - 3abc$

Also taught in the form of:

$(a + b + c)(\frac{(a - b)^2 + (b - c)^2 + (c - a)^2}{2}) = a^3 + b^3 + c^3 - 3abc$

To prove, expand the given expression. That's kinda painful, if you ask me, but do it once, or maybe thrice to satisfy any doubts you have.

Two implications of the above equation are:

1. If a + b + c = 0, then $a^3 + b^3 + c^3 = 3abc$.

2. If a = b = c, then $a^3 + b^3 + c^3 = 3abc$.

A third implication is the converse of the above:

3. If $a^3 + b^3 + c^3 = 3abc$, then either a + b + c = 0, or a = b = c.