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PUMaC 2010 · 代数（A 组） · 第 5 题

PUMaC 2010 — Algebra (Division A) — Problem 5

专题
Discrete Math / 离散数学
难度
L3
来源
PUMaC

题目详情

  1. Let f ( x ) = 3 x − 5 x + 2 x − 6. If the roots of f are given by α , β , and γ , find ( ) ( ) ( ) 2 2 2 1 1 1
    • . α − 2 β − 2 γ − 2
解析
  1. Let f ( x ) = 3 x − 5 x + 2 x − 6. If the roots of f are given by α , β , and γ , find ( ) ( ) ( ) 2 2 2 1 1 1
    • . α − 2 β − 2 γ − 2 Solution: 68. A polynomial with roots α − 2, β − 2, and γ − 2 is given by 3 2 g ( x ) = f ( x + 2) = 3 x + 13 x + 18 x + 2 . A polynomial with roots 1 / ( α − 2), 1 / ( β − 2), and 1 / ( γ − 2) is given by 3 2 h ( x ) = 2 x + 18 x + 13 x + 3 . 2 2 2 2 Since a + b + c = ( a + b + c ) − 2( ab + bc + ca ), we can find the result by finding the elementary symmetric polynomials on the roots. Here, we have 1 1 1
    • = − 9 α − 2 β − 2 γ − 2 and 1 1 1 1 1 1 13
    • = , α − 2 β − 2 β − 2 γ − 2 γ − 2 α − 2 2 13 2 so the desired sum is ( − 9) − 2 = 81 − 13 = 68. 2