HMMT 十一月 2015 · 冲刺赛 · 第 22 题

英文解析

22. [ 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).
    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 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 ithat 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