Rational Root Test

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Problem 58 (Rational Root Test)

Suppose that $$f(x) = a_nx^n+\cdots +a_1x+a_0\in \mathbb{Z}[x],$$ with $a_n\neq 0$. Prove that if $r$ and $s$ are relatively prime and $f(r/s)=0$, then we must have $r\mid a_0$ and $s\mid a_n$.

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