Overview

The symmetric cubic identity appears in both algebra and number theory. It is especially powerful when $x+y+z=0$.

Key Ideas

• $x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2-xy-xz-yz)$.
• If $x+y+z=0$, then $x^3+y^3+z^3=3xyz$.
• Use it to convert a cubic sum into quadratic symmetric terms.

Core Skills

Check for $x+y+z=0$

If the sum is zero, the identity collapses immediately and saves a lot of work.

Convert to Symmetric Sums

Rewrite the right-hand factor as $(x^2+y^2+z^2)-(xy+xz+yz)$ to connect with known sums.

Factor the Expression

Use the identity to factor $x^3+y^3+z^3-3xyz$ and analyze when it is zero.

Worked Example

If $x+y+z=0$ and $xyz=4$, compute $x^3+y^3+z^3$.

By the special case, $x^3+y^3+z^3=3xyz$. So the value is $3\cdot 4 = 12$.

More Examples

Example 1: Evaluate a Sum

If $x+y+z=0$ and $x^2+y^2+z^2=18$, find $x^3+y^3+z^3$.

Since $x^3+y^3+z^3=3xyz$, find $xyz$ from $(x+y+z)^2 = x^2+y^2+z^2+2(xy+xz+yz)$ to get $xy+xz+yz=-9$. Then use $(x+y+z)(x^2+y^2+z^2-xy-xz-yz)=x^3+y^3+z^3-3xyz$ to solve.

Example 2: Factorization

Factor $a^3+b^3+c^3-3abc$.

It equals $(a+b+c)(a^2+b^2+c^2-ab-ac-bc)$.

Example 3: When Is It Zero?

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

Strategy Checklist

• Check if $x+y+z=0$ to simplify.
• Express $xy+xz+yz$ using $(x+y+z)^2$ if needed.
• Factor first before expanding.

Common Pitfalls

• Assuming $x^3+y^3+z^3$ is determined by $x+y+z$ alone.
• Forgetting the sign of the $3xyz$ term.
• Expanding instead of using the identity directly.

Practice Problems