For the same quartic, which expression equals αβ + αγ + αδ + βγ + βδ + γδ?

For the same quartic, which expression equals αβ + αγ + αδ + βγ + βδ + γδ?

• A. -d/a
• B. c/a
• C. e/a
• D. -b/a

Answer: (B)

The key idea is that the coefficients of a polynomial encode sums of products of its roots (Vieta’s formulas). For a quartic ax^4 + bx^3 + cx^2 + dx + e with roots α, β, γ, δ, the sum of all pairwise products is αβ + αγ + αδ + βγ + βδ + γδ = c/a. This is exactly the expression in question, so it equals c/a. The other standard symmetric sums come from the other coefficients: the sum of the roots is -b/a, the triple products sum is -d/a, and the product of all four roots is e/a, which is why the other options don’t match.

The key idea is that the coefficients of a polynomial encode sums of products of its roots (Vieta’s formulas). For a quartic ax^4 + bx^3 + cx^2 + dx + e with roots α, β, γ, δ, the sum of all pairwise products is αβ + αγ + αδ + βγ + βδ + γδ = c/a. This is exactly the expression in question, so it equals c/a. The other standard symmetric sums come from the other coefficients: the sum of the roots is -b/a, the triple products sum is -d/a, and the product of all four roots is e/a, which is why the other options don’t match.