jmc

Question

A cubic polynomial f(x) = x^3 + ax^2 + bx + c with at least two distinct roots has the following properties: (i) The sum of all the roots is equal to twice the product of all the roots. (ii) The sum of the squares of all the roots is equal to 3 times the product of all the roots. (iii) f(1) = 1. Find c.

Accepted Answer

Let r, s, t be the root of the cubic. Then by Vieta's formulas, align* r + s + t &= -a, rs + rt + st &= b, rst &= -c. align*From condition (i), -a = -2c, so a = 2c. Squaring the equation r + s + t = -a, we get r^2 + s^2 + t^2 + 2(rs + rt + st) = a^2.Then r^2 + s^2 + t^2 = a^2 - 2(rs + rt + st) = a^2 - 2b.Then from condition (ii), a^2 - 2b = -3c, so b = a^2 + 3c2 = 4c^2 + 3c2.Finally, from condition (iii), f(1) = 1 + a + b + c = 1, so a + b + c = 0. Substituting, we get 2c + 4c^2 + 3c2 + c = 0.This simplifies to 4c^2 + 9c = 0. Then c(4c + 9) = 0, so c = 0 or c = -94. If c = 0, then a = b = 0, which violates the condition that f(x) have at least two distinct roots. Therefore, c = -94.