If the quartic is monic (a = 1), the sum of pairwise products αβ + αγ + αδ + βγ + βδ + γδ equals which expression?

If the quartic is monic (a = 1), the sum of pairwise products αβ + αγ + αδ + βγ + βδ + γδ equals which expression?

• A. -d
• B. c
• C. -b
• D. e

Answer: (B)

Vieta’s formulas tell us how the coefficients of a polynomial relate to sums and products of its roots. For a monic quartic with roots α, β, γ, δ, you can write the polynomial as (x − α)(x − β)(x − γ)(x − δ). Expanding this shows that the x^3 term has coefficient equal to the negative of the sum of the roots, the x^2 term has as its coefficient the sum of all pairwise products αβ + αγ + αδ + βγ + βδ + γδ, the x term has coefficient minus the sum of triple products, and the constant term is the product of all four roots. If the polynomial is written as x^4 + b x^3 + c x^2 + d x + e, these comparisons give: - sum of the roots = −b - sum of pairwise products = c - sum of triple products = −d - product of the roots = e Therefore, the sum of all pairwise products αβ + αγ + αδ + βγ + βδ + γδ is exactly the coefficient c.

Vieta’s formulas tell us how the coefficients of a polynomial relate to sums and products of its roots. For a monic quartic with roots α, β, γ, δ, you can write the polynomial as (x − α)(x − β)(x − γ)(x − δ). Expanding this shows that the x^3 term has coefficient equal to the negative of the sum of the roots, the x^2 term has as its coefficient the sum of all pairwise products αβ + αγ + αδ + βγ + βδ + γδ, the x term has coefficient minus the sum of triple products, and the constant term is the product of all four roots.

If the polynomial is written as x^4 + b x^3 + c x^2 + d x + e, these comparisons give:

• sum of the roots = −b

• sum of pairwise products = c

• sum of triple products = −d

• product of the roots = e

Therefore, the sum of all pairwise products αβ + αγ + αδ + βγ + βδ + γδ is exactly the coefficient c.