jmc

Question

Suppose that there exist nonzero complex numbers a, b, c, and d such that k is a root of both the equations ax^3 + bx^2 + cx + d = 0 and bx^3 + cx^2 + dx + a = 0. Enter all possible values of k, separated by commas.

Accepted Answer

We have that align* ak^3 + bk^2 + ck + d &= 0, bk^3 + ck^2 + dk + a &= 0. align*Multiplying the first equation by k, we get ak^4 + bk^3 + ck^2 + dk = 0.Subtracting the equation bk^3 + ck^2 + dk + a = 0, we get ak^4 = a. Since a is nonzero, k^4 = 1. Then k^4 - 1 = 0, which factors as (k - 1)(k + 1)(k^2 + 1) = 0.This means k is one of 1, -1, i, or -i. If a = b = c = d = 1, then -1, i, and -i are roots of both polynomials. If a = b = c = 1 and d = -3, then 1 is a root of both polynomials. Therefore, the possible values of k are 1,-1,i,-i.