Determine all possible values of the expression A^3 + B^3 + C^3 - 3ABC, where A, B, and C are nonnegative integers. 

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.

Explore the values of the expression A^3 + B^3 + C^3 - 3ABC using algebraic identities and factorization techniques. Understand how nonnegative integers A, B, and C influence the outcome. Suitable for Grades 9-12 students studying algebra and number theory.

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