Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 127

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\[x^{2} - 3x - 4 = 0\]

\[x_{1} + x_{2} = 3;\ \ \ x_{1} \cdot x_{2} = - 4.\]

\[1)\ \frac{1^{\backslash x_{2}}}{x_{1}} + \frac{1^{\backslash x_{1}}}{x_{2}} = \frac{x_{2} + x_{1}}{x_{1} \cdot x_{2}} =\]

\[= \frac{3}{- 4} = - 0,75.\]

\[2)\ x_{1}^{2} + x_{2}^{2} = x_{1}^{2} + 2x_{1}x_{2} + x_{2}^{2} -\]

\[- 2x_{1}x_{2} = \left( x_{1} + x_{2} \right)^{2} - 2x_{1}x_{2} =\]

\[= 3^{2} - 2 \cdot ( - 4) = 9 + 8 = 17.\]

\[3)\ x_{1}^{3} + x_{2}^{3} =\]

\[= \left( x_{1} + x_{2} \right)\left( x_{1}^{2} - x_{1}x_{2} + x_{2}^{2} \right) =\]

\[= 3 \cdot (17 + 4) = 3 \cdot 21 = 63.\ \]