Determination of All Possible Values of the Expression A3B3C3 - 3ABC

The expression A3B3C3 - 3ABC, where A, B, and C are nonnegative integers, can be analyzed using algebraic identities to determine all possible values it can take. Let's explore this in detail.

Steps to Analyze the Expression

Nonnegative Integers: Since A, B, and C are nonnegative integers, the term A3B3C3 - 3ABC is also a nonnegative integer.

Factorization: The expression can be factored using the sum of cubes identity:

A3B3C3 - 3ABC (ABC) (ABC2 - A - BC - AC) - 3ABC (ABC) (A2B2C2 - AB - AC - BC)

Second Factor Analysis

The term (A2B2C2 - AB - AC - BC) can be rewritten using the identity:

A2B2C2 - AB - AC - BC (frac{1}{2} ) (A - B)2 (B - C)2 (C - A)2

This expression is always nonnegative since it is a sum of squares.

Possible Values of the Expression

If A B C: The expression A3B3C3 - 3ABC 0.

If any two of A, B, C are equal and the third is different: The expression will yield nonnegative values.

If all three values are distinct: The expression will still yield nonnegative values.

Specific Cases and Values

Case 1: A B C 0 gives 0.

Case 2: A 1, B 1, C 1 gives 0.

Case 3: A 1, B 1, C 0 gives 1.

Case 4: A 2, B 2, C 0 gives 8 - 0 8.

Conclusion: As A, B, and C take on nonnegative integer values, the expression A3B3C3 - 3ABC can take on all nonnegative integer values. Specifically, it can take values that correspond to the distinct sums of cubes and linear combinations of these integers, leading to a variety of outputs based on the specific values of A, B, and C.

Thus, the possible values of the expression include 0, 1, 2, 3, 4, etc., depending on the combinations of A, B, and C. The exact values can be computed for specific integers but in general, the expression can yield a wide range of nonnegative integers.