Which expression equals α^3 + β^3 + γ^3 in terms of α+β+γ and αβ+βγ+αγ and αβγ?

Which expression equals α^3 + β^3 + γ^3 in terms of α+β+γ and αβ+βγ+αγ and αβγ?

• A. (α+β+γ)^3 - 3(αβ+βγ+αγ)(α+β+γ) + 3αβγ
• B. (α+β+γ)^3 - 3αβ(α+β+γ)
• C. α^3 + β^3 + γ^3
• D. (α+β+γ)^3 - 3(αβ+βγ+αγ)(α+β+γ)

Answer: (A)

The expression α^3 + β^3 + γ^3 can be tied to the sums S1 = α + β + γ, S2 = αβ + βγ + αγ, and S3 = αβγ through the identity α^3 + β^3 + γ^3 − 3αβγ = (α + β + γ)(α^2 + β^2 + γ^2 − αβ − βγ − αγ). Replace α^2 + β^2 + γ^2 with (α + β + γ)^2 − 2(αβ + βγ + αγ). This gives α^3 + β^3 + γ^3 − 3αβγ = (α + β + γ)[(α + β + γ)^2 − 3(αβ + βγ + αγ)] = (α + β + γ)^3 − 3(αβ + βγ + αγ)(α + β + γ). Now add 3αβγ to both sides to isolate α^3 + β^3 + γ^3: α^3 + β^3 + γ^3 = (α + β + γ)^3 − 3(αβ + βγ + αγ)(α + β + γ) + 3αβγ. This matches the given expression, so that form is the correct one. The other options miss the +3αβγ term or present an incorrect combination.

The expression α^3 + β^3 + γ^3 can be tied to the sums S1 = α + β + γ, S2 = αβ + βγ + αγ, and S3 = αβγ through the identity α^3 + β^3 + γ^3 − 3αβγ = (α + β + γ)(α^2 + β^2 + γ^2 − αβ − βγ − αγ). Replace α^2 + β^2 + γ^2 with (α + β + γ)^2 − 2(αβ + βγ + αγ). This gives

α^3 + β^3 + γ^3 − 3αβγ = (α + β + γ)[(α + β + γ)^2 − 3(αβ + βγ + αγ)]

= (α + β + γ)^3 − 3(αβ + βγ + αγ)(α + β + γ).

Now add 3αβγ to both sides to isolate α^3 + β^3 + γ^3:

α^3 + β^3 + γ^3 = (α + β + γ)^3 − 3(αβ + βγ + αγ)(α + β + γ) + 3αβγ.

This matches the given expression, so that form is the correct one. The other options miss the +3αβγ term or present an incorrect combination.