Question: Given that \(a_{n}, a_{n-1}, \ldots, a_{1}, a_{0}\) are all integers and that \(a_{n}\) and \(a_{0}\) are nonzero, derive a necessary condition for the equation \(a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}=0\) to have a rational root. Then use this condition to prove that \(\sqrt{n-1}+\sqrt{n+1}\) is irrational for every integer \(n \geq 1\)

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