ГДЗ по алгебре 8 класс Макарычев Задание 238

\[\boxed{\text{238\ (238).\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[\textbf{а)}\ \frac{5x}{(x - 2)(x + 3)} =\]

\[= \frac{a^{\backslash x + 3}}{x - 2} + \frac{b^{\backslash x - 2}}{x + 3}\]

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

\[= \frac{ax + 3a + bx - 2b}{(x - 2)(x + 3)}\]

\[5x = ax + bx + 3a - 2b\]

\[5x + 0 = x(a + b) + (3a - 2b)\]

\[\left\{ \begin{matrix} a + b = 5\ \ \ \ \ \\ 3a - 2b = 0 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} a = 5 - b\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 3(5 - b) - 2b = 0 \\ \end{matrix} \right.\ \]

\[15 - 3b - 2b = 0\]

\[15 - 5b = 0\]

\[b = 3.\]

\[a = 5 - b = 5 - 3 = 2.\]

\[Ответ:a = 2,\ b = 3.\]

\[\textbf{б)}\frac{5x + 31}{(x - 5)(x + 2)} =\]

\[= \frac{a^{\backslash x + 2}}{x - 5} - \frac{b^{\backslash x - 5}}{x + 2}\]

\[\frac{5x + 31}{(x - 5)(x + 2)} =\]

\[= \frac{ax + 2a - bx + 5b}{(x - 5)(x + 2)}\]

\[5x + 31 = x(a - b) + (2a + 5b)\]

\[\left\{ \begin{matrix} a - b = 5\ \ \ \ \ \ \ \ \\ 2a + 5b = 31 \\ \end{matrix} \right.\ \ \]

\[a = 5 + b\]

\[2(5 + b) + 5b = 31\]

\[10 + 2b + 5b = 31\]

\[7b = 21\]

\[b = 3.\]

\[a = 5 + b = 5 + 3 = 8.\]

\[Ответ:a = 8,\ b = 3.\]