Symmetric Cubic Of Roots

Symmetric power sums of roots are computed from the sum and product via algebraic identities, a routine JEE Advanced technique avoiding explicit roots. For x^2 - 2x + 4 = 0, Vieta gives alpha + beta = 2 and alpha beta = 4. The identity alpha^3 + beta^3 = (alpha + beta)^3 - 3 alpha beta (alpha + beta) applies. Substituting, (2)^3 - 3·4·2 = 8 - 24 = -16. The cube-sum identity expresses the answer entirely through the elementary symmetric functions, the standard pattern. Option 8 uses only the first term (alpha + beta)^3. Option 16 drops the sign of the correction. Option -8 miscomputes the product term. Hence alpha^3 + beta^3 = -16. Plausibility check: the roots are 1 ± i√3, complex conjugates whose cubes are conjugate, so their sum is real, and computing (1 + i√3)^3 = -8 gives -8 + -8 = -16, independently confirming the identity-based result. Power sums of roots build systematically from the elementary symmetric functions through Newton's identities, of which the cube-sum formula used here is a special case. Because the roots form a conjugate pair, their cubes are also conjugates and their sum is necessarily real, a structural check that independently corroborates the value obtained purely from the sum and product via the algebraic identity.