PUMaC 2009 · 代数（A 组） · 第 1 题

PUMaC 2009 — Algebra (Division A) — Problem 1

Find the root that the following three polynomials have in common: 3 2 x + 41 x − 49 x − 2009 3 2 x + 5 x − 49 x − 245 3 2 x + 39 x − 117 x − 1435

解析

Find the root that the following three polynomials have in common: 3 2 x + 41 x − 49 x − 2009 3 2 x + 5 x − 49 x − 245 3 2 x + 39 x − 117 x − 1435 (Hint: use all three polynomials.) Solution. 7. Since the answer to each question on this test is integer-valued, we are looking for integer solutions. It is known that an integer root of a monic polynomial having integer coefficients must divide the constant coefficient. In our case the common root must divide 2 GCD (2009 , 1435 , 245). It is easy to factor 245 = 5 × 7 and check that 7 divides both 2009 and 1435, but 49 does not divide 435, and 5 does not divide 2009. Hence the GCD is 7, so the common roots (if any) must be ± 1 or ± 7. One can check that only 7 is a common root.