Which expression gives α^3 + β^3 + γ^3 + δ^3 in terms of sums and products?

Which expression gives α^3 + β^3 + γ^3 + δ^3 in terms of sums and products?

• A. (α+β+γ+δ)^3 - 3(α+β+γ+δ)(αβ+αγ+αδ+βγ+βδ+γδ) + 3(αβγ + αβδ + αγδ + βγδ)
• B. (α+β+γ+δ)^3 - 3(αβ+βγ+γδ+αδ)(α+β+γ+δ)
• C. α^3 + β^3 + γ^3 + δ^3
• D. (α+β+γ+δ)^3 - 3(αβ+βγ+αγ)(α+β+γ+δ)

Answer: (A)

Think in terms of symmetric sums: S1 is the sum α+β+γ+δ, S2 is the sum of all pairwise products αβ+αγ+αδ+βγ+βδ+γδ, and S3 is the sum of all triple products αβγ+αβδ+αγδ+βγδ. If you expand (α+β+γ+δ)^3, you get the sum of the cubes α^3+β^3+γ^3+δ^3 plus various cross terms. Those cross terms can be collected into two groups: terms with a square times another variable and terms that are products of three distinct variables. When you subtract 3 times the product of the sum with the sum of pairwise products, you cancel the square-type cross terms. The remaining triple-term cross terms combine to give 3 times the sum of triple products. What’s left is exactly α^3+β^3+γ^3+δ^3, so you obtain α^3+β^3+γ^3+δ^3 = (α+β+γ+δ)^3 − 3(α+β+γ+δ)(αβ+αγ+αδ+βγ+βδ+γδ) + 3(αβγ+αβδ+αγδ+βγδ). This matches the expression that uses the sums S1, S2, and S3, which is why it’s the correct form. The other options either omit necessary cross-term structures or simply restate the left-hand side without expressing it in terms of the required sums and products.

Think in terms of symmetric sums: S1 is the sum α+β+γ+δ, S2 is the sum of all pairwise products αβ+αγ+αδ+βγ+βδ+γδ, and S3 is the sum of all triple products αβγ+αβδ+αγδ+βγδ. If you expand (α+β+γ+δ)^3, you get the sum of the cubes α^3+β^3+γ^3+δ^3 plus various cross terms. Those cross terms can be collected into two groups: terms with a square times another variable and terms that are products of three distinct variables. When you subtract 3 times the product of the sum with the sum of pairwise products, you cancel the square-type cross terms. The remaining triple-term cross terms combine to give 3 times the sum of triple products. What’s left is exactly α^3+β^3+γ^3+δ^3, so you obtain

α^3+β^3+γ^3+δ^3 = (α+β+γ+δ)^3 − 3(α+β+γ+δ)(αβ+αγ+αδ+βγ+βδ+γδ) + 3(αβγ+αβδ+αγδ+βγδ).

This matches the expression that uses the sums S1, S2, and S3, which is why it’s the correct form. The other options either omit necessary cross-term structures or simply restate the left-hand side without expressing it in terms of the required sums and products.