HMMT 十一月 2015 · 冲刺赛 · 第 22 题

HMMT November 2015 — Guts Round — Problem 22

解析

[ 12 ] Find all the roots of the polynomial x − 5 x + 11 x − 13 x + 9 x − 3. Proposed by: Sam Korsky √ √ √ √ 3+ 3 i 1 − 3 i 3 − 3 i 1+ 3 i Answer: 1 , , , , 2 2 2 2 5 4 5 3 2 The x − 5 x at the beginning of the polynomial motivates us to write it as ( x − 1) + x − 3 x +4 x − 2 and 3 2 5 3 again the presence of the x − 3 x motivates writing the polynomial in the form ( x − 1) +( x − 1) +( x − 1). 2 Let a and b be the roots of the polynomial x + x + 1. It’s clear that the roots of our polynomial are 2 2 given by 1 and the roots of the polynomials ( x − 1) = a and ( x − 1) = b . The quadratic formula shows √ √ √ √ − 1+ 3 i − 1 − 3 i 1+ 3 i 1 − 3 i that WLOG a = and b = so we find that either x − 1 = ± or x − 1 = ± . 2 2 2 2 √ √ √ √ 3 + 3 i 1 − 3 i 3 − 3 i 1 + 3 i Hence our roots are 1 , , , , . 2 2 2 2