Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 266

\[\boxed{\mathbf{266}.}\]

\[1)\ 3xy + 10x - 13y - 35 =\]

\[= 0\ \ \ | \cdot 3\]

\[9xy - 39y + 30x - 105 = 0\]

\[9xy - 39y + 30x -\]

\[- 130 + 25 = 0\]

\[3y(3x - 13) +\]

\[+ 10 \cdot (3x - 13) = - 25\]

\[(3x - 13)(3y + 10) = - 25\]

\[Делители:\ \pm 1;\ \pm 5;\ \pm 25.\]

\[\left\{ \begin{matrix} 3x - 13 = 1\ \ \ \ \ \\ 3y + 10 = - 25 \\ \end{matrix} \right.\ \text{\ \ \ }\left\{ \begin{matrix} 3x = 14\ \ \ \\ 3y = - 35 \\ \end{matrix} \right.\ \text{\ \ }\]

\[\ \left\{ \begin{matrix} x = \frac{14}{3}\text{\ \ \ \ } \\ y = - \frac{35}{3} \\ \end{matrix} \right.\ \Longrightarrow не\ целые;\]

\[\left\{ \begin{matrix} 3x - 13 = - 1 \\ 3y + 10 = 25 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\left\{ \begin{matrix} 3x = 12 \\ 3y = 15 \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\left\{ \begin{matrix} x = 4 \\ y = 5 \\ \end{matrix} \right.\ \Longrightarrow (4;5);\]

\[\left\{ \begin{matrix} 3x - 13 = 25\ \ \\ 3y + 10 = - 1 \\ \end{matrix} \right.\ \text{\ \ \ \ }\left\{ \begin{matrix} 3x = 38\ \ \ \\ 3y = - 11 \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\left\{ \begin{matrix} x = \frac{38}{3}\text{\ \ \ \ } \\ y = - \frac{11}{3} \\ \end{matrix} \right.\ \Longrightarrow не\ целые;\]

\[\left\{ \begin{matrix} 3x - 13 = - 25 \\ 3y + 10 = 1\ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\left\{ \begin{matrix} 3x = - 12 \\ 3y = - 9\ \ \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\left\{ \begin{matrix} x = - 4 \\ y = - 3 \\ \end{matrix} \right.\ \Longrightarrow ( - 4;\ - 3);\]

\[\left\{ \begin{matrix} 3x - 13 = 5\ \ \ \\ 3y + 10 = - 5 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\left\{ \begin{matrix} 3x = 18\ \ \ \ \\ 3y = - 15 \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\ \left\{ \begin{matrix} x = 6\ \ \ \\ y = - 5 \\ \end{matrix} \right.\ \Longrightarrow (6;\ - 5);\]

\[\left\{ \begin{matrix} 3x - 13 = - 5 \\ 3y + 10 = 5\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\left\{ \begin{matrix} 3x = 8\ \ \ \\ 3y = - 5 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = \frac{8}{3}\text{\ \ \ \ } \\ y = - \frac{5}{3} \\ \end{matrix} \right.\ \Longrightarrow не\ целые.\]

\[Ответ:(4;5);( - 4;\ - 3);(6; - 5).\]

\[2)\ 3xy + 19x + 10y + 55 =\]

\[= 0\ \ \ | \cdot 3\]

\[9xy + 57x + 30y + 165 = 0\]

\[9xy + 30y + 57x +\]

\[+ 190 - 25 = 0\]

\[3y(3x + 10) +\]

\[+ 19(3x + 10) = 25\]

\[(3x + 10)(3y + 19) = 25\]

\[Делители:\ \pm 1;\ \pm 5;\ \pm 25.\]

\[\left\{ \begin{matrix} 3x + 10 = 25 \\ 3y + 19 = 1\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ }\left\{ \begin{matrix} 3x = 15\ \ \ \ \\ 3y = - 18 \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = 5\ \ \ \\ y = - 6 \\ \end{matrix} \right.\ \Longrightarrow (5;\ - 6);\]

\[\left\{ \begin{matrix} 3x + 10 = 1\ \ \\ 3y + 19 = 25 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\left\{ \begin{matrix} 3x = - 9 \\ 3y = 6\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\ \left\{ \ \begin{matrix} x = - 3 \\ y = 2\ \ \ \\ \end{matrix} \right.\ \Longrightarrow ( - 3;2);\ \ \ \ \]

\[\left\{ \begin{matrix} 3x + 10 = - 1\ \ \\ 3y + 19 = - 25 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} 3x = - 11 \\ 3y = - 44 \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\text{\ \ }\left\{ \begin{matrix} x = - \frac{11}{3} \\ y = - \frac{44}{3} \\ \end{matrix} \right.\ \Longrightarrow не\ целые;\]

\[\left\{ \begin{matrix} 3x + 10 = - 25 \\ 3y + 19 = - 1\ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} 3x = - 35 \\ 3y = - 20 \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\left\{ \begin{matrix} x = - \frac{35}{3} \\ y = - \frac{20}{3} \\ \end{matrix} \right.\ \Longrightarrow не\ целые;\]

\[\left\{ \begin{matrix} 3x + 10 = 5 \\ 3y + 19 = 5 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\left\{ \begin{matrix} 3x = - 5\ \ \\ 3y = - 14 \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = - \frac{5}{3}\text{\ \ \ } \\ y = - \frac{14}{3} \\ \end{matrix} \right.\ \Longrightarrow не\ целые;\]

\[\left\{ \begin{matrix} 3x + 10 = - 5 \\ 3y + 19 = - 5 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\left\{ \begin{matrix} 3x = - 15 \\ 3y = - 24 \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\ \left\{ \begin{matrix} x = - 5 \\ y = - 8 \\ \end{matrix} \right.\ \Longrightarrow ( - 5;\ - 8).\]

\[Ответ:( - 3;2);(5; - 6);\]

\[( - 5;\ - 8).\]

\[3)\ x^{3} - 6x^{2} - xy + 13x +\]

\[+ 3y + 7 = 0\]

\[x^{3} - 3x^{2} - xy + 3y - 3x^{2} +\]

\[+ 9x + 3x - 9 + 9 + x - 3 +\]

\[+ 3 + 7 = 0\]

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

\[- 3x(x - 3) + 3(x - 3) +\]

\[+ (x - 3) + 19 = 0\]

\[(x - 3)\left( x^{2} - y - 3x + 4 \right) = - 19\]

\[Делители:\ \pm 1; \pm 19.\]

\[\left\{ \begin{matrix} x - 3 = \ 19\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - y - 3x + 4 = - 1\ \\ \end{matrix} \right.\ \text{\ \ }\]

\[\left\{ \begin{matrix} x = 22\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y = 22^{2} - 3 \cdot 22 + 5 \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\left\{ \begin{matrix} x = 22\ \ \\ y = 423 \\ \end{matrix} \right.\ \Longrightarrow (22;423);\]

\[\left\{ \begin{matrix} x - 3 = - 19\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - y - 3x + 4 = 1 \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\left\{ \begin{matrix} x = - 16\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y = ( - 16)^{2} + 3 \cdot 16 + 3 \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\left\{ \begin{matrix} x = - 16 \\ y = 307\ \\ \end{matrix} \right.\ \Longrightarrow ( - 16;307);\]

\[\left\{ \begin{matrix} x - 3 = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - y - 3x + 4 = - 19 \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\ \left\{ \begin{matrix} x = 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y = 4^{2} - 12 + 23 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 4\ \ \ \\ y = 27 \\ \end{matrix} \right.\ \Longrightarrow (4;27);\]

\[\left\{ \begin{matrix} x - 3 = - 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - y - 3x + 4 = 19\ \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y = 2^{2} - 6 - 15 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 2\ \ \ \ \ \ \\ y = - 17 \\ \end{matrix} \right.\ \Longrightarrow (2; - 17).\]

\[Ответ:(22;423);( - 16;307);\]

\[(4;27);(2; - 17).\]

\[4)\ x^{3} - xy - 7x + 2y + 23 = 0\]

\[x^{3} - 2x^{2} + 2x^{2} - xy + 2y -\]

\[- 4x - 3x + 6 + 17 = 0\]

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

\[- y(x - 2) - 3(x - 2) + 17 = 0\]

\[(x - 2)\left( x^{2} + 2x - y - 3 \right) = - 17\]

\[Делители:\ \pm 1;\ \pm 17.\]

\[\left\{ \begin{matrix} x - 2 = 17\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} + 2x - y - 3 = - 1 \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\ \left\{ \begin{matrix} x = 19\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y = 19^{2} + 38 - 2 \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\left\{ \begin{matrix} x = 19\ \ \\ y = 397 \\ \end{matrix} \right.\ \Longrightarrow (19;397);\]

\[\left\{ \begin{matrix} x - 2 = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} + 2x - y - 3 = - 17 \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\left\{ \begin{matrix} x = 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y = 3^{2} + 6 + 14 \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\ \left\{ \begin{matrix} x = 3\ \ \ \\ y = 29 \\ \end{matrix} \right.\ \Longrightarrow (3;29);\]

\[\left\{ \begin{matrix} x - 2 = - 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} + 2x - y - 3 = 17 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y = 1 + 2 - 20 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x = 1\ \ \ \ \ \\ y = - 17 \\ \end{matrix} \right.\ \Longrightarrow (1;\ - 17);\]

\[\left\{ \begin{matrix} x - 2 = - 17\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} + 2x - y - 3 = 1\ \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\ \left\{ \begin{matrix} x = - 15\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ y = ( - 15)^{2} - 30 - 4 \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\left\{ \begin{matrix} x = - 15 \\ y = 191\ \\ \end{matrix} \right.\ \Longrightarrow ( - 15;191).\]

\[Ответ:(3;29);\ \ (1; - 17);\ \]

\[\ (19;397);\ \ ( - 15;191).\]