PUMaC 2016 · 代数（B 组） · 第 3 题

PUMaC 2016 — Algebra (Division B) — Problem 3

Bob draws the graph of y = x − 13 x + 40 x + 25 and is dismayed to find out that it only has one root. Alice comes to the rescue, translating (without rotating or dilating) the axes so that the origin is at the point that used to be ( − 20 , 16). This new graph has three x -intercepts; compute their sum.

解析

The translation of the axes is equivalent to translating the polynomial 20 to the right and 16 down. Thus, the new polynomial has the equation 3 2 y = ( x − 20) − 13( x − 20) + 40( x − 20) + 25 − 16 . 2 The sum of the roots of this polynomial is negative the coefficient of x , which is − 3 · 20 − 13 = − 73. Thus, the answer is 73 . Alternatively, note that changing the constant coefficient does not change the sum of the roots, and then translating the polynomial 20 to the right increases the real part of each root by 20. Thus, the sum of the roots goes from 13 to 13 + 60 = 73. Problem written by Eric Neyman.