HMMT 二月 2010 · GEN2 赛 · 第 5 题

英文解析

5. [ 5 ] 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 (includingcomplex values).
   Answer: 1, − 1, i , − i Let k be a root of both polynomials. Multiplying the first polynomial by kand subtracting the second, we have ak − a = 0, which means that k is either 1, − 1, i , or − i . If 4
   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 .