The sum of the triple products αβγ + αβδ + αγδ + βγδ equals which expression in terms of a and d?

The sum of the triple products αβγ + αβδ + αγδ + βγδ equals which expression in terms of a and d?

• A. -b/a
• B. c/a
• C. -d/a
• D. e/a

Answer: (C)

The concept being used is Vieta’s formulas for a polynomial with roots α, β, γ, δ. For a quartic with leading coefficient a, written as a x^4 + b x^3 + c x^2 + d x + e, dividing by a gives the monic form x^4 + (b/a)x^3 + (c/a)x^2 + (d/a)x + (e/a). In the expanded form (x−α)(x−β)(x−γ)(x−δ), the coefficient of x is minus the sum of all triple products αβγ + αβδ + αγδ + βγδ. Therefore the sum of these triple products equals the negative of the coefficient of x in the monic polynomial, which is −d/a. So the sum αβγ + αβδ + αγδ + βγδ equals −d/a. The other coefficients relate to the other symmetric sums: the sum of the roots is −b/a, the sum of pairwise products is c/a, and the product of all four roots is e/a.

The concept being used is Vieta’s formulas for a polynomial with roots α, β, γ, δ. For a quartic with leading coefficient a, written as a x^4 + b x^3 + c x^2 + d x + e, dividing by a gives the monic form x^4 + (b/a)x^3 + (c/a)x^2 + (d/a)x + (e/a). In the expanded form (x−α)(x−β)(x−γ)(x−δ), the coefficient of x is minus the sum of all triple products αβγ + αβδ + αγδ + βγδ. Therefore the sum of these triple products equals the negative of the coefficient of x in the monic polynomial, which is −d/a.

So the sum αβγ + αβδ + αγδ + βγδ equals −d/a. The other coefficients relate to the other symmetric sums: the sum of the roots is −b/a, the sum of pairwise products is c/a, and the product of all four roots is e/a.