A condition connecting variables can simplify an expression before any individual variable is known. The key is to use an identity with a valid derivation, rather than assume that taking a power preserves addition.
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State and verify the factorisation
The relevant identity is a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca).
Expanding the right-hand side cancels the mixed squared terms and leaves the stated cubic expression. This algebraic identity is valid before any zero-sum condition is imposed.
Apply the condition exactly where it belongs
If a + b + c = 0, the first factor on the right vanishes. Consequently a3 + b3 + c3 = 3abc.
For the original numerical example a = 2, b = 3 and c = -5, the cube sum is 8 + 27 - 125 = -90. The product expression gives 3 × 2 × 3 × (-5) = -90 as well.
Do not cube a sum term by term
Although (a + b + c)3 is zero under the condition, it does not follow that a3 + b3 + c3 is zero. Expanding a cube of a sum produces cross terms.
The numerical example is a direct counterexample to that tempting shortcut: its three variables sum to zero, but their cubes sum to minus ninety.
Know what the relation can and cannot determine
If the sum is zero and abc is known, the identity determines the cube sum immediately. If only the zero sum is known, the cube sum need not have one fixed numerical value because the product can vary.
This is an original AMC 12 learning exercise, not an official problem. A strong review includes both the factorisation and a counterexample to the incorrect rule, so the shortcut remains attached to its precise condition.
Continue reading: AMC 12 Sequences: Derive a Geometric Sum by Subtracting Two Lines.
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Official information and further reading: Source 1
