PUMaC 2012 · 代数（A 组） · 第 4 题

PUMaC 2012 — Algebra (Division A) — Problem 4

[ 4 ] Let f be a polynomial of degree 3 with integer coefficients such that f (0) = 3 and f (1) = 11. If f has exactly 2 integer roots, how many such polynomials f exist?

解析

[ 4 ] Let f be a polynomial of degree 3 with integer coefficients such that f (0) = 3 and f (1) = 11. If f has exactly 2 integer roots, how many such polynomials f exist? Solution: The answer is 0, but the argument is more general: if f (0) and f (1) are odd, then we claim that f can’t have any integer roots: Suppose a is an integer solution. Then f ( x ) = ( x − a ) g ( x ), and g ( x ) also has integer coefficients. So f (0) = − ag (0) and f (1) = (1 − a ) g (1), where g (0) and g (1) are also integers. Since either a or 1 − a is even, f (0) and f (1) can’t be both odd, as in the hypothesis, so f has no integer roots. Answer: 0