Symmetric‑Sum Identity for Cubics

Problem.

Real numbers $x, y, z$ satisfy

$$ x+y+z = 6, \qquad xy+yz+zx = 11, \qquad xyz = 6. $$

Without solving for $x, y, z$ individually, find the value of

$$ x^{3}+y^{3}+z^{3}. $$

One‑Minute Solution (use the cubic symmetric identity)

A standard identity for any three variables is

Plug in the given symmetric sums:

$$ \begin{aligned} x^{3}+y^{3}+z^{3} &= 6^{3}

• 3(6)(11)

• 3(6) \[4pt] &= 216

• 198

• 18 \[4pt] &= 36. \end{aligned} $$

$$ \boxed{36} $$