Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 338

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\[x + y = 1;\ \ xy = - 2:\]

\[(x + y)^{2} = x^{2} + 2xy + y^{2} =\]

\[= x^{2} + y^{2} + 2 \cdot ( - 2) = 1\]

\[x^{2} + y^{2} = 5;\ \ \left( \text{xy} \right)^{2} = 4.\]

\[x^{6} + y^{6} = \left( x^{2} \right)^{3} + \left( y^{2} \right)^{3} =\]

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

\[= 5 \cdot \left( \left( x^{2} \right)^{2} + 2\left( \text{xy} \right)^{2} + \left( y^{2} \right)^{2} - 2\left( \text{xy} \right)^{2} - \left( \text{xy} \right)^{2} \right) =\]

\[= 5 \cdot \left( \left( x^{2} + y^{2} \right)^{2} - 3\left( \text{xy} \right)^{2} \right) =\]

\[= 5 \cdot (25 - 3 \cdot 4) = 5 \cdot 13 = 65.\]

\[Ответ:65.\]