If the quartic is monic, the sum of triple products αβγ + αβδ + αγδ + βγδ equals which expression?

If the quartic is monic, the sum of triple products αβγ + αβδ + αγδ + βγδ equals which expression?

• A. -b
• B. c
• C. -d
• D. e

Answer: (C)

Think of the quartic as built from its roots α, β, γ, δ: f(x) = (x−α)(x−β)(x−γ)(x−δ). When you expand this, the signs of the elementary symmetric sums alternate: x^4 − (sum of roots) x^3 + (sum of pairwise products) x^2 − (sum of triple products) x + (product of roots). So the coefficient of x is minus the sum of the triple products αβγ + αβδ + αγδ + βγδ. If the polynomial is written as x^4 + b x^3 + c x^2 + d x + e, then −(sum of triple products) = d, hence the sum of triple products equals −d.

Think of the quartic as built from its roots α, β, γ, δ: f(x) = (x−α)(x−β)(x−γ)(x−δ). When you expand this, the signs of the elementary symmetric sums alternate: x^4 − (sum of roots) x^3 + (sum of pairwise products) x^2 − (sum of triple products) x + (product of roots).

So the coefficient of x is minus the sum of the triple products αβγ + αβδ + αγδ + βγδ. If the polynomial is written as x^4 + b x^3 + c x^2 + d x + e, then −(sum of triple products) = d, hence the sum of triple products equals −d.