HMMT 二月 2013 · 冲刺赛 · 第 20 题

英文解析

20. [ 11 ] The polynomial f ( x ) = x − 3 x − 4 x + 4 has three real roots r , r , and r . Let g ( x ) =
    1 2 3
    3 2 2
    x + ax + bx + c be the polynomial which has roots s , s , and s , where s = r + r z + r z ,
    1 2 3 1 1 2 3
    √
    − 1+ i 3
    2 2
    s = r z + r z + r , s = r z + r + r z , and z = . Find the real part of the sum of the
    2 1 2 3 3 1 2 3
    coefficients of g ( x ).2
    2 π
    2 π 2 πi 3 2
    Answer: − 26 Note that z = e = cos + i sin , so that z = 1 and z + z + 1 = 0. Also,3
    3 3
    s = s z and s = s z .2
    2 1 3 1
    Then, the sum of the coefficients of g ( x ) is g (1) = (1 − s )(1 − s )(1 − s ) = (1 − s )(1 − s z )(1 − s z ) =2
    1 2 3 1 1 1
    2 2 3 2 3 3 3
    1 − (1 + z + z ) s + ( z + z + z ) s − z s = 1 − s .
    1 1 11
    3 2 3 3 3 3 2 2 2 2 2 2 2
    Meanwhile, s = ( r + r z + r z ) = r + r + r + 3 r r z + 3 r r z + 3 r r z + 3 r r z + 3 r r z +
    1 2 3 2 3 3 1 1
    1 1 2 3 1 1 2 2 3
    2 2
    3 r r z + 6 r r r .
    2 1 2 3
    2 313
    Since the real parts of both z and z are − , and since all of r , r , and r are real, the real part of s is
    1 2 3
    3 9 2721
    3 3 3 2 2 3
    r + r + r − ( r r + · · · + r r )+6 r r r = ( r + r + r ) − ( r + r + r )( r r + r r + r r )+ r r r =
    2 2 1 2 3 1 2 3 1 2 3 1 2 2 3 3 1 1 2 3
    1 2 3 1 3
    2 2 2
    9 27
    3 − · 3 · − 4 + · − 4 = 27.3
    2 2
    Therefore, the answer is 1 − 27 = − 26.
    2013