Question

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

Answer 1

Prove the statement .

Given: $a+b+c=0$

Thus, $a+b=-c$…………………… $\left(1\right)$

Cubing both sides :

$\Rightarrow$ ${\left(a+b\right)}^3=-c^3$

$\Rightarrow$$a^3+b^3+3ab(a+b)=-c^3$

$\Rightarrow$ $a^3+b^3+c^3=-3ab(a+b)$

From equation $1$:

$\Rightarrow$ $a^3+b^3+c^3=3abc$ ($a+b=-c$)

Hence proved that $a^3+b^3+c^3=3abc$ .