Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 251

\[\boxed{\mathbf{251}.}\]

\[a = \frac{n^{4} - n^{3} + 2n^{2}}{n^{2} + 1} =\]

\[= \frac{n^{4} + n^{2} - n^{3} - n + n + n^{2}}{n^{2} + 1} =\]

\[= \frac{n^{2}\left( n^{2} + 1 \right)}{n^{2} + 1} - \frac{n\left( n^{2} + 1 \right)}{n^{2} + 1} +\]

\[+ \frac{n + n^{2}}{n^{2} + 1} =\]

\[= n^{2} - n + \frac{n^{2} + n}{n^{2} + 1};\ \ n \in Z;\]

\[\text{\ \ }n^{2} \in Z.\]

\[n = - 1:\]

\[\frac{- 1 + ( - 1)^{2}}{( - 1)^{2} + 1} = 0.\]

\[n = 0:\]

\[\frac{1 + 1}{0 + 1} = 0.\]

\[n = 1:\]

\[\frac{1 + 1^{2}}{1^{2} + 1} = 1.\]

\[n = 2:\]

\[\frac{2 + 2^{2}}{2^{2} + 1} = \frac{6}{5} \notin Z.\]

\[Ответ:\ - 1;\ \ 1;\ \ 0.\]