If the quartic is monic, the product αβγδ equals which expression?

If the quartic is monic, the product αβγδ equals which expression?

• A. e
• B. a
• C. -d
• D. e/a

Answer: (D)

Vieta's formulas tell us how the roots of a polynomial relate to its coefficients. For a quartic written as a x^4 + b x^3 + c x^2 + d x + e = 0 with roots α, β, γ, δ, you can factor it as a(x−α)(x−β)(x−γ)(x−δ)=0. Expanding shows the constant term is a times the product of the roots, so e = a αβγδ. Rearranging gives αβγδ = e/a. Note: if the quartic were monic (leading coefficient 1), the product would be e/1 = e. The given form uses leading coefficient a, hence the product is e/a.

Vieta's formulas tell us how the roots of a polynomial relate to its coefficients. For a quartic written as a x^4 + b x^3 + c x^2 + d x + e = 0 with roots α, β, γ, δ, you can factor it as a(x−α)(x−β)(x−γ)(x−δ)=0. Expanding shows the constant term is a times the product of the roots, so e = a αβγδ. Rearranging gives αβγδ = e/a.

Note: if the quartic were monic (leading coefficient 1), the product would be e/1 = e. The given form uses leading coefficient a, hence the product is e/a.