For the same quartic, which expression equals αβγδ?

For the same quartic, which expression equals αβγδ?

• A. -b/a
• B. c/a
• C. -d/a
• D. e/a

Answer: (D)

The expression for the product of all four roots comes from Viète’s formulas for a quartic. If the quartic is ax^4 + bx^3 + cx^2 + dx + e with roots α, β, γ, δ, then it can be written as a(x−α)(x−β)(x−γ)(x−δ). The constant term of this expanded form is a(−α)(−β)(−γ)(−δ) = aαβγδ, which must equal e. Therefore the product of the roots is αβγδ = e/a. So the expression that represents the product of all four roots is e/a. The other results correspond to the sum of the roots (−b/a), the sum of pairwise products (c/a), and the sum of triple products (−d/a).

The expression for the product of all four roots comes from Viète’s formulas for a quartic. If the quartic is ax^4 + bx^3 + cx^2 + dx + e with roots α, β, γ, δ, then it can be written as a(x−α)(x−β)(x−γ)(x−δ). The constant term of this expanded form is a(−α)(−β)(−γ)(−δ) = aαβγδ, which must equal e. Therefore the product of the roots is αβγδ = e/a.

So the expression that represents the product of all four roots is e/a. The other results correspond to the sum of the roots (−b/a), the sum of pairwise products (c/a), and the sum of triple products (−d/a).