HMMT 二月 1998 · 代数 · 第 7 题

HMMT February 1998 — Algebra — Problem 7

Given that three roots of f ( x ) = x + ax + bx + c are 2, -3, and 5, what is the value of a + b + c ? x + 1 3 x + 4 9

解析

Problem: Given that three roots of f ( x ) = x + ax + bx + c are 2, -3, and 5, what is the value of a + b + c ? 3 3 Solution: By definition, the coefficient of x is negative the sum of the roots. In f ( x ), the coefficient of x is 0. Thus the sum of the roots of f ( x ) is 0. Then the fourth root is -4. Then f ( x ) = ( x − 2)( x +3)( x − 5)( x +4). Notice that f (1) is 1 + a + b + c . Thus our answer is f (1) − 1 = (1 − 2)(1 + 3)(1 − 5)(1 + 4) − 1 = 79 . x +1 3 x +4 9