HMMT 二月 2001 · 代数 · 第 7 题
HMMT February 2001 — Algebra — Problem 7
题目详情
英文原题
- A polynomial P has four roots, , , 2 , 4. The product of the roots is 1, and P (1) = 1.
4 2
Find P (0).
解析
英文解析
- A polynomial P has four roots, , , 2 , 4. The product of the roots is 1, and P (1) = 1.
4 2
Find P (0).
Solution: A polynomial Q with n roots, x , . . . , x , and Q ( x ) = 1 is given by Q ( x ) =
1 n 0
( x − x )( x − x ) ··· ( x − x )
1 2 n 1 8
, so P (0) = = .
3 1
( x − x )( x − x ) ··· ( x − x ) 9
· · ( − 1) · ( − 3)
0 1 0 2 0 4
4 2