To determine all possible values of the expression $A^3 + B^3 + C^3 - 3ABC$, where $A$, $B$, and $C$ are nonnegative integers, we can utilize a useful algebraic identity.

The expression $A^3 + B^3 + C^3 - 3ABC$ can be factored using the identity:

$A^3 + B^3 + C^3 - 3ABC = (A+B+C)(A^2 + B^2 + C^2 - AB - BC - CA)$

Analysis of the Identity

1. If $A + B + C = 0$:

   • Since $A$, $B$, and $C$ are nonnegative integers, the only way this can happen is if $A = B = C = 0$.
   • Substituting $A = B = C = 0$ into the expression, we get: $0^3 + 0^3 + 0^3 - 3 \cdot 0 \cdot 0 \cdot 0 = 0$
   • Hence, 0 is a possible value.

2. If $A + B + C > 0$:

   • In this case, $(A+B+C)$ is a positive integer.
   • The term $(A^2 + B^2 + C^2 - AB - BC - CA)$ represents a nonnegative integer because it is equivalent to: $\frac{1}{2} \left( (A-B)^2 + (B-C)^2 + (C-A)^2 \right)$
   • Thus, the entire product $(A+B+C)(A^2 + B^2 + C^2 - AB - BC - CA)$ is a nonnegative integer.

Observations

• The identity shows that for any nonnegative integers $A$, $B$, and $C$, the expression $A^3 + B^3 + C^3 - 3ABC$ simplifies to a nonnegative integer.
• As demonstrated, $A^3 + B^3 + C^3 - 3ABC$ can be zero if $A = B = C = 0$.
• For other nonnegative integer values of $A$, $B$, and $C$, the expression will be some nonnegative integer based on the specific values of $A$, $B$, and $C$.

Conclusion

The possible values of the expression $A^3 + B^3 + C^3 - 3ABC$ are all nonnegative integers, including zero.

Would you like further details or have any specific questions?

Here are 5 more questions you might find interesting:

1. What are the roots of the polynomial $x^3 - 6x^2 + 11x - 6$?
2. How do you solve a system of linear equations using matrix methods?
3. Can you explain the method of completing the square for quadratic equations?
4. What is the significance of the determinant of a matrix?
5. How do you find the derivative of a composite function using the chain rule?

Tip: When tackling algebraic identities, try to recognize common factorization patterns which can simplify solving the equations.