Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 340

\[\boxed{\mathbf{340}.}\]

\[x^{3} - 2x^{2} - x + 2 = 0\]

\[x_{1};x_{2};x_{2} - корни.\]

\[\left\{ \begin{matrix} x_{1} + x_{2} + x_{3} = 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x_{1}x_{2} + x_{2}x_{3} + x_{1}x_{3} = - 1 \\ x_{1}x_{2}x_{3} = 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[x_{1} = - y_{1};\ \ x_{2} = - y_{2};\ \ x_{3} = - y_{3}\]

\[Подставим:\]

\[1)\ y_{1} + y_{2} + y_{3} =\]

\[= - \left( x_{1} + x_{2} + x_{3} \right) = - 2;\]

\[2)\ y_{1}y_{2} + y_{2}y_{3} + y_{1}y_{3} =\]

\[= x_{1}x_{2} + x_{2}x_{3} + x_{1}x_{3} = - 1;\]

\[3)\ y_{1}y_{2}y_{3} = - x_{1}x_{2}x_{3} = - 2.\]

\[Получаем\ уравнение:\]

\[x^{3} + 2x^{2} - x - 2 = 0.\]