Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 291

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\[1)\ 2x^{5} + 4x^{2} + 5x - 3 =\]

\[= (x - 1)(2x^{4} + ax^{3} + bx^{2} - 2x + 3)\]

\[(x - 1)\left( 2x^{4} + ax^{3} + bx^{2} - 2x + 3 \right) =\]

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

\[+ bx^{3} - bx^{2} - 2x^{2} + 2x +\]

\[+ 3x - 3 =\]

\[= 2x^{5} + (a - 2)x^{4} +\]

\[+ (b - a)x^{3} + ( - b - 2)x^{2} +\]

\[+ 5x - 3\]

\[\left\{ \begin{matrix} a - 2 = 0\ \ \ \ \ \ \\ b - a = 0\ \ \ \ \ \ \\ - b - 2 = - 4 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\left\{ \begin{matrix} a = 2\ \ \ \ \ \ \ \\ b = a\ \ \ \ \ \ \\ - b = - a \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} a = 2 \\ b = 2 \\ \end{matrix} \right.\ \]

\[Ответ:a = 2;\ \ b = 2.\]

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

\[= (x - 1)(x^{4} - ax^{3} + 2x^{2} + 2x + b)\]

\[(x - 1)\left( x^{4} - ax^{3} + 2x^{2} + 2x + b \right) =\]

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

\[- 2x^{2} - 2x + bx - b =\]

\[= x^{5} + ( - 1 - a)x^{4} +\]

\[+ (a + 2)x^{3} + (b - 2)x - b\]

\[\left\{ \begin{matrix} - 1 - a = 0 \\ a + 2 = 1\ \ \ \\ b - 2 = 0\ \ \ \\ - b = - 2\ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\left\{ \begin{matrix} a = - 1 \\ b = 2\ \ \ \\ \end{matrix} \right.\ \]

\[Ответ:a = - 1;\ \ b = 2.\]