HMMT 二月 2014 · 冲刺赛 · 第 12 题

英文解析

12. [ 6 ] Find a nonzero monic polynomial P ( x ) with integer coefficients and minimal degree such that
    √ √
    3 3
    P (1 − 2 + 4) = 0. (A polynomial is called monic if its leading coefficient is 1.)
    √ √ √ √ √
    3 3 3 3 3
    3 23
    √
    Answer: x − 3 x + 9 x − 9 Note that (1 − 2 + 4)(1 + 2) = 3, so 1 − 2 + 4 = .
    1+ 23
    √
    3 3 3 23
    Now, if f ( x ) = x − 2, we have f ( 2) = 0, so if we let g ( x ) = f ( x − 1) = ( x − 1) − 2 = x − 3 x +3 x − 3,
    √ √ √
    3 3 3
    3 27 27 9 3
    √
    then g (1+ 2) = f ( 2) = 0. Finally, we let h ( x ) = g ( ) = − + − 3 so h ( ) = g (1+ 2) = 0.
    3 2 3
    x x x x
    1+ 2
    x 3
    3 2
    To make this a monic polynomial, we multiply h ( x ) by − to get x − 3 x + 9 x − 9. 3