Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 386

\[\boxed{\mathbf{386}.}\]

\[1)\ 9x^{3} + 12x^{2} - 10x + 4 = 0\]

\[Делители:\ \pm 1;\ \pm 2;\ \pm 4.\]

\[P( - 2) = 0 \rightarrow x = - 2\ (корень).\]

\[P(x) =\]

\[= (x + 2)\left( 9x^{2} - 6x + 2 \right) = 0\]

\[9x^{2} - 6x + 2 = 0\]

\[D_{1} = 9 - 18 = - 8 < 0\]

\[нет\ корней.\]

\[Ответ:x = - 2.\]

\[2)\ x^{4} + x^{3} - 5x^{2} + x - 6 = 0\]

\[Делители:\ \pm 1;\ \pm 2;\ \pm 3;\ \pm 6.\]

\[P(2) = 0;\ \ P( - 3) = 0;\]

\[x = - 3;\ \ x = 2 - корни.\]

\[(x - 2)(x + 3) = x^{2} + x - 6.\]

\[P(x) =\]

\[= (x - 2)(x + 3)\left( x^{2} + 1 \right) = 0\]

\[x^{2} + 1 = 0\]

\[x^{2} = - 1\]

\[нет\ корней.\]

\[Ответ:x = 2;\ - 3.\]

\[3)\ x^{5} + 3x^{4} + 2x^{3} +\]

\[+ 6x^{2} + 2x + 6 = 0\]

\[Делители:\ \pm 1; \pm 2; \pm 3;\ \pm 6.\]

\[x^{4}(x + 3) + 2x^{2}(x + 3) +\]

\[+ 2(x + 3) = 0\]

\[(x + 3)\underset{\neq 0}{\overset{\left( x^{4} + 2x^{2} + 2 \right)}{︸}} = 0\]

\[Ответ:x = - 3.\]

\[4)\ x^{5} - 2x^{4} - 3x^{3} + 6x^{2} -\]

\[- 4x + 8 = 0\]

\[x^{4}(x - 2) - 3x^{2}(x - 2) -\]

\[- 4 \cdot (x - 2) = 0\]

\[(x - 2)\left( x^{4} - 3x^{2} - 4 \right) = 0\]

\[x^{4} - 3x^{2} - 4 = 0\]

\[x^{2} = t \geq 0:\]

\[t^{2} - 3t - 4 = 0\]

\[t_{1} + t_{2} = 3;\ \ t_{1} \cdot t_{2} = - 4\]

\[t_{1} = 4;\ \ \ \ \]

\[t_{2} = - 1\ (не\ подходит).\]

\[x^{2} = 4\]

\[x = \pm 2.\]

\[Ответ:x = \pm 2.\]