ГДЗ по алгебре и начала математического анализа 10 класс Колягин Задание 293

\[\boxed{\mathbf{293}.}\]

\[1)\ R(x) = ax + b - остаток:\]

\[x^{7} + x^{6} - 6x^{5} + x^{2} - 5x =\]

\[+ \left( x^{2} + x - 6 \right) \cdot M(x) + ax + b\]

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

\[x_{1} + x_{2} = - 1;\ \ x_{1} \cdot x_{2} = - 6\]

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

\[При\ x = 2:\]

\[0 \cdot M(x) + 2a + b = 128 +\]

\[+ 64 - 192 + 4 - 10\]

\[2a + b = - 6\]

\[b = - 6 - 2a.\]

\[При\ x = - 3:\]

\[0 \cdot M(x) - 3a + b = - 2187 +\]

\[+ 729 + 1458 + 9 + 15\]

\[- 3a + b = 24\]

\[- 3a - 6 - 2a = 24\]

\[- 5a = 30\]

\[a = - 6.\]

\[b = - 6 + 12 = 6.\]

\[Ответ:R(x) = - 6x + 6.\]

\[2)\ R(x) = ax + b - остаток:\]

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

\[= \left( x^{2} - 4 \right) \cdot M(x) + ax + b\]

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

\[x^{2} = 4\]

\[x = \pm 2.\]

\[При\ x = 2:\]

\[0 \cdot M(x) + 2a + b =\]

\[= 32 - 32 + 8 + 2 - 2\]

\[2a + b = 8\]

\[b = 8 - 2a.\]

\[При\ x = - 2:\]

\[0 \cdot M(x) - 2a + b = - 32 -\]

\[- 32 - 8 - 8 - 2 - 2\]

\[- 2a + b = - 76.\]

\[- 2a + 8 - 2a = - 76\]

\[- 4a = - 84\]

\[a = 21.\]

\[b = 8 - 2 \cdot 21 = 8 - 42 = - 34.\]

\[Ответ:R(x) = 21x - 34.\]