HMMT 二月 2010 · 代数 · 第 4 题

HMMT February 2010 — Algebra — Problem 4

[ 4 ] Suppose that there exist nonzero complex numbers a , b , c , and d such that k is a root of both the 3 2 3 2 equations ax + bx + cx + d = 0 and bx + cx + dx + a = 0. Find all possible values of k (including complex values). 20 20

解析

[ 4 ] Suppose that there exist nonzero complex numbers a , b , c , and d such that k is a root of both the 3 2 3 2 equations ax + bx + cx + d = 0 and bx + cx + dx + a = 0. Find all possible values of k (including complex values). Answer: 1, − 1, i , − i Let k be a root of both polynomials. Multiplying the first polynomial by k 4 and subtracting the second, we have ak − a = 0, which means that k is either 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. So k can be 1, − 1, i , and − i . 20 20