HMMT 二月 2009 · 代数 · 第 5 题

英文解析

5. [ 4 ] Let a , b , and c be the 3 roots of x − x + 1 = 0. Find + + .
   a +1 b +1 c +1
   Answer: − 2
   Solution: We can substitute x = y − 1 to obtain a polynomial having roots a + 1, b + 1, c + 1, namely,
   3 3 2
   ( y − 1) − ( y − 1) + 1 = y − 3 y + 2 y + 1. The sum of the reciprocals of the roots of this polynomialis, by Viete’s formulas, = − 2.2 − 1 π