For the same quartic, which expression equals αβγ + αβδ + αγδ + βγδ?

For the same quartic, which expression equals αβγ + αβδ + αγδ + βγδ?

• A. -b/a
• B. e/a
• C. -d/a
• D. c/a

Answer: (C)

Vieta’s formulas connect the roots to the coefficients of the quartic. If the quartic is a(x−α)(x−β)(x−γ)(x−δ) and you expand it, the coefficient of x is -a times the sum of all triple products αβγ + αβδ + αγδ + βγδ. Since this coefficient must equal d, you get d = -a(αβγ + αβδ + αγδ + βγδ). Rearranging gives αβγ + αβδ + αγδ + βγδ = -d/a. So the expression in question equals -d/a. This aligns with the general pattern: sum of roots is -b/a, sum of pairwise products is c/a, the sum of triple products is -d/a, and the product of all roots is e/a.

Vieta’s formulas connect the roots to the coefficients of the quartic. If the quartic is a(x−α)(x−β)(x−γ)(x−δ) and you expand it, the coefficient of x is -a times the sum of all triple products αβγ + αβδ + αγδ + βγδ. Since this coefficient must equal d, you get d = -a(αβγ + αβδ + αγδ + βγδ). Rearranging gives αβγ + αβδ + αγδ + βγδ = -d/a. So the expression in question equals -d/a. This aligns with the general pattern: sum of roots is -b/a, sum of pairwise products is c/a, the sum of triple products is -d/a, and the product of all roots is e/a.