HMMT 二月 2014 · 代数 · 第 4 题

英文解析

4. Let b and c be real numbers, and define the polynomial P ( x ) = x + bx + c . Suppose that P ( P (1)) =
   P ( P (2)) = 0, and that P (1) 6 = P (2). Find P (0).
   3 1
   Answer: − OR − 1 . 5 OR − 1 Since P ( P (1)) = P ( P (2)) = 0, but P (1) 6 = P (2), it follows that
   2 2
   P (1) = 1 + b + c and P (2) = 4 + 2 b + c are the distinct roots of the polynomial P ( x ). Thus, P ( x )
   factors:
   P ( x ) = x + bx + c 2 = ( x − (1 + b + c ))( x − (4 + 2 b + c )) = x − (5 + 3 b + 2 c ) x + (1 + b + c )(4 + 2 b + c ) .2
   It follows that − (5 + 3 b + 2 c ) = b , and that c = (1 + b + c )(4 + 2 b + c ). From the first equation, wefind 2 b + c = − 5 / 2. Plugging in c = − 5 / 2 − 2 b into the second equation yields − 5 / 2 − 2 b = (1 + ( − 5 / 2) − b )(4 + ( − 5 / 2)) .
   1 3
   Solving this equation yields b = − , so c = − 5 / 2 − 2 b = − .
   2 2
   4 3 2