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刚好答过差不多的一道题, 这里再贴一下.设f(x) = x^m+x-1, g(x) = x^n+x^2-1.设多项式带余除法f(x) = g(x)q(x)+r(x), 余式r(x)为0或次数小于g(x)的次数.由带余除法的步骤, 这里的q(x)与r(x)都是有理系数多项式, 乘以适当正整数k后为整系数多项式.∵k·f(x)/g(x) = k·q(x)+k·r(x)/g(x)在无穷多个整数上取整值, 而k·q(x)总取整值,∴k·r(x)/g(x)在无穷多个整数上取整值.若r(x)非零, 由其次数小于g(x), 对|x|充分大, 总有0 < |k·r(x)| < |g(x)|, 比值不为整数.至多只有有限个x使其为整数, ... |
