proof of rational root theoremLet p(x)∈ℤ[x]. Let n be a positive integer with degp(x)=n. Let c0,…,cn∈ℤ such that p(x)=cnxn+cn-1xn-1+⋯+c1x+c0.Let a,b∈ℤ with gcd(a,b)=1 and b>0 such that ab is a root of p(x). Then0=p(ab)=cn(ab)n+cn-1(ab)n-1+⋯+c1⋅ab+c0=cn⋅anbn+cn-1⋅an-1bn-1+⋯+c1⋅ab+c0.Multiplying through by bn and rearranging yields:cnan+cn-1an-1b+⋯+c1abn-1+c0bn=0c0bn=-cnan-cn-1an-1b-⋯-c1abn-1c0bn=a(-cnan-1-cn-1an-2b-⋯-c1bn-1)Thus, a|c0bn and, by hypothesis, gcd(a,b)=1. This implies that a|c0.Similarly:cnan+cn-1an-1b+⋯+c1abn-1+c0bn=0cnan=-cn-1an-1b-⋯-c1abn-1-c0bncnan=b(-cn-1an-1-⋯-c1abn-1-c0bn-1)Therefore, b|cnan and b|cn.