HMMT 十一月 2011 · 冲刺赛 · 第 21 题

英文解析

21. [ 10 ] Let P ( x ) = x + 2 x − 13 x − 14 x + 24 be a polynomial with roots r , r , r , r . Let Q be the
    1 2 3 4
    2 2 2 2 4
    quartic polynomial with roots r , r , r , r , such that the coefficient of the x term of Q is 1. Simplify
    1 2 3 4
    the quotient Q ( x ) /P ( x ), leaving your answer in terms of x . (You may assume that x is not equal to 2
    any of r , r , r , r ).
    1 2 3 4
    4 3 2
    Answer: x − 2 x − 13 x + 14 x + 24 We note that we must have
    2 2 2 2 2 2 2 2 2 2 2 2 2
    Q ( x ) = ( x − r )( x − r )( x − r )( x − r ) ⇒ Q ( x ) = ( x − r )( x − r )( x − r )( x − r )
    1 2 3 4 1 2 3 4
    . Since P ( x ) = ( x − r )( x − r )( x − r )( x − r ) , we get that
    1 2 3 4
    Q ( x ) /P ( x ) = ( x + r )( x + r )( x + r )( x + r ) .2
    1 2 3 4
    2 4
    Thus, Q ( x ) /P ( x ) = ( − 1) P ( − x ) = P ( − x ), so it follows that
    2 4 3 2
    Q ( x ) /P ( x ) = x − 2 x − 13 x + 14 x + 24 .