Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 393

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\[P(x) = x^{6} + 6x^{5} + cx^{4}.\]

\[1)\ \ x^{6} + 6x^{5} + cx^{4}\ \vdots x + 2\]

\[P( - 2) = 64 - 32b + 16c = 0;\]

\[P(3) = 729 + 243b + 81c = 0.\]

\[\left\{ \begin{matrix} 64 - 32b + 16c = 0\ \ \ \ \ \ \ |\ :16 \\ 729 + 243b + 81c = 0\ \ \ |\ :81 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} - 2b + c = - 4 \\ 3b + c = - 9\ \ \\ \end{matrix} \right.\ ( - )\]

\[- 5b = 5\]

\[b = - 1.\]

\[c = 2b - 4 = - 2 - 4 = - 6.\]

\[Ответ:b = - 1;\ \ c = - 6.\]

\[2)\ P(x)\ \vdots (x - 4)\]

\[P(4) = 4^{6} + 4^{5}b + 4^{4}c = 0;\]

\[P(x)\ \vdots (x + 5)\]

\[P( - 5) = ( - 5)^{6} + ( - 5)^{5}b +\]

\[+ ( - 5)^{4}c = 0.\]

\[\left\{ \begin{matrix} 4096 + 1024b + 256c = 0\ \ \ \ \ |\ :256 \\ 15\ 625 - 3125b + 625c = 0\ \ |\ :625 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 4b + c = - 16\ \ \\ - 5b + c = - 25 \\ \end{matrix} \right.\ ( - )\]

\[9b = 9\]

\[b = 1.\]

\[c = 5b - 25 = 5 - 25 = - 20.\]

\[Ответ:b = 1;\ \ c = - 25.\]