Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 311

\[\boxed{\mathbf{311}.}\]

\[P(x) = M_{1}(x)(x + 4) + 5\]

\[P(x) = M_{2}(x)(x - 5) + 14\]

\[P(x) = M_{3}(x)(x + 4)(x - 5) +\]

\[+ ax + b\]

\[x = - 4:\]

\[P( - 4) = 0 \cdot M_{1}(x) + 5 = 5;\]

\[P( - 4) = 0 \cdot M_{3}(x) - 4a + b =\]

\[= - 4a + b.\]

\[x = 5:\]

\[P(5) = 0 \cdot M_{2}(x) + 14 = 14;\]

\[P(5) = 0 \cdot M_{3}(x) + 5a + b =\]

\[= 5a + b.\]

\[\left\{ \begin{matrix} - 4a + b = 5 \\ 5a + b = 14\ \\ \end{matrix} \right.\ ( - )\]

\[- 9a = - 9\]

\[a = 1.\]

\[b = 5 + 4a = 5 + 4 = 9.\]

\[Ответ:\ \ x + 9.\]