α^3 + β^3 + γ^3 equals which expression?

α^3 + β^3 + γ^3 equals which expression?

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

Answer: (A)

The key idea is to relate the sum of cubes to the cube of the sum and the symmetric sums. If you expand the cube of the sum s = α + β + γ, you get s^3 = α^3 + β^3 + γ^3 + 3s(αβ + βγ + αγ) - 3αβγ. Rearranging this to solve for α^3 + β^3 + γ^3 gives α^3 + β^3 + γ^3 = s^3 - 3s(αβ + βγ + αγ) + 3αβγ, which matches the expression (α + β + γ)^3 - 3(α + β + γ)(αβ + βγ + αγ) + 3αβγ. This is the correct form because it directly comes from expanding the cube and collecting like terms. The other options don’t align with this rearrangement: one misses the +3αβγ term, another doubles or misplaces the product terms, and one just restates the left-hand side.

The key idea is to relate the sum of cubes to the cube of the sum and the symmetric sums. If you expand the cube of the sum s = α + β + γ, you get

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

Rearranging this to solve for α^3 + β^3 + γ^3 gives

α^3 + β^3 + γ^3 = s^3 - 3s(αβ + βγ + αγ) + 3αβγ,

which matches the expression (α + β + γ)^3 - 3(α + β + γ)(αβ + βγ + αγ) + 3αβγ. This is the correct form because it directly comes from expanding the cube and collecting like terms. The other options don’t align with this rearrangement: one misses the +3αβγ term, another doubles or misplaces the product terms, and one just restates the left-hand side.