Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 344

\[\boxed{\mathbf{344}.}\]

\[a\left( x^{4} + y^{4} + x^{2}y^{2} \right) +\]

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

\[делится\ на\ (x + y)(2x - y).\]

\[при\ x = - y:\]

\[P(x;y) = 3ax^{4} + x^{4} = 0;\]

\[при\ y = 2x:\]

\[P(x;y) = 21ax^{4} -\]

\[- 6bx^{4} + 16x^{4} = 0.\]

\[\left\{ \begin{matrix} 3a + 1 = 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 21a - 6b + 16 = 0 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]

\[3a = 1\]

\[a = - \frac{1}{3}.\]

\[6b = 21a + 16 =\]

\[= 21 \cdot \left( - \frac{1}{3} \right) + 16 =\]

\[= - 7 + 16 = 9\]

\[b = 9\ :6 = 1,5.\]

\[Ответ:a = - \frac{1}{3};\ \ \ b = 1,5.\]