### Theory:

Identity VIII: $$a^3 + b^3 + c^3 -3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ac)$$

Let us verify the above identity by direct multiplication of the right hand side expression.

$$(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ac)$$ $$=$$ $$a (a^2 + b^2 + c^2 - ab - bc - ac) + b(a^2 + b^2 + c^2 - ab - bc - ac) + c(a^2 + b^2 + c^2 - ab - bc - ac)$$

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

Simplify the above expression by combining the like terms.

$$=$$ $$a^3 + b^3 + c^3 - 3abc$$

Thus we have the identity $$a^3 + b^3 + c^3 -3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ac)$$.
Example:
Expand $$27x^3 + 8y^3 + z^3 - 18xyz$$.

Solution:

Let us write the expression of $$27x^3 + 8y^3 + z^3 - 18xyz$$ using the identity ${a}^{3}+{b}^{3}+{c}^{3}-3\mathit{abc}=\left(a+b+c\right)\left({a}^{2}+{b}^{2}+{c}^{2}-\mathit{ab}-\mathit{bc}-\mathit{ac}\right)$.

${27x}^{3}+{8y}^{3}+{z}^{3}-12\mathit{xyz}={\left(3x\right)}^{3}+{\left(2y\right)}^{3}+{\left(z\right)}^{3}-3\left(3x\right)\left(2y\right)\left(z\right)$

$$=$$ $\left(3x+2y+z\right)\phantom{\rule{0.147em}{0ex}}\left({9x}^{2}+{4y}^{2}+{z}^{2}-\left(3x\right)\left(2y\right)-\left(2y\right)\left(z\right)-\left(3x\right)\left(z\right)\right)$

$$=$$ $\left(3x+2y+z\right)\left({3x}^{2}+{2y}^{2}-6\mathit{xy}-2\mathit{yz}-3\mathit{xz}\right)$
Important!
If $$a+b+c = 0$$ then the identity ${a}^{3}+{b}^{3}+{c}^{3}-3\mathit{abc}=\left(a+b+c\right)\left({a}^{2}+{b}^{2}+{c}^{2}-\mathit{ab}-\mathit{bc}-\mathit{ac}\right)$ is rewritten as follows:

${a}^{3}+{b}^{3}+{c}^{3}-3\mathit{abc}=\left(0\right)\left({a}^{2}+{b}^{2}+{c}^{2}-\mathit{ab}-\mathit{bc}-\mathit{ac}\right)$

${a}^{3}+{b}^{3}+{c}^{3}-3\mathit{abc}=\left(0\right)$

${a}^{3}+{b}^{3}+{c}^{3}=3\mathit{abc}$
Example:
Evaluate $$8^3 - 5^3 - 3^3$$.

Solution:

By the identity if $a+b+c=0$, then ${a}^{3}+{b}^{3}+{c}^{3}=3\mathit{abc}$.

On comparing $$8^3 - 5^3 - 3^3$$ with $a+b+c=0$ we have $$a$$ $$=$$ $$8$$, $$b$$ $$=$$ $$-5$$ and $$c$$ $$=$$ $$-3$$.

Here $$a + b + c$$ $$=$$ $$8 - 5 - 3$$ $$=$$ $$0$$

Therefore, $$8^3 - 5^3 - 3^3$$ $$=$$ $$3 \times 8 \times -5 \times -3$$

$$=$$ $$360$$