Question: Given that \({{a}_{n}},{{a}_{n-1}},....,{{a}_{1}},{{a}_{0}}\) are all integers and that \({{a}_{n}}\) and \({{a}_{0}}\) are non-zero, derive a necessary condition for the equation \({{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+...+{{a}_{1}}x+{{a}_{0}}=0\), to have a rational root. Then use this condition to prove that \(\sqrt{n-1}+\sqrt{n+1}\).