Let α, β, γ, δ be the roots of the quartic a x^4 + b x^3 + c x^2 + d x + e = 0, with a ≠ 0. Which expression equals α + β + γ + δ?

Let α, β, γ, δ be the roots of the quartic a x^4 + b x^3 + c x^2 + d x + e = 0, with a ≠ 0. Which expression equals α + β + γ + δ?

• A. -b/a
• B. b/a
• C. -d/a
• D. d/a

Answer: (A)

Consider the polynomial written with its roots α, β, γ, δ: a(x−α)(x−β)(x−γ)(x−δ). When you expand, the x^4 term is a, and the x^3 term comes from picking one linear factor contributing x and the others contributing constants, giving −a(α+β+γ+δ) as the coefficient of x^3. Since the original polynomial has coefficient b on x^3, we have b = −a(α+β+γ+δ). Therefore α+β+γ+δ = −b/a. The other expressions correspond to sums of products of the roots taken two or three at a time, not the total sum.

Consider the polynomial written with its roots α, β, γ, δ: a(x−α)(x−β)(x−γ)(x−δ). When you expand, the x^4 term is a, and the x^3 term comes from picking one linear factor contributing x and the others contributing constants, giving −a(α+β+γ+δ) as the coefficient of x^3. Since the original polynomial has coefficient b on x^3, we have b = −a(α+β+γ+δ). Therefore α+β+γ+δ = −b/a. The other expressions correspond to sums of products of the roots taken two or three at a time, not the total sum.