Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 504

\[\boxed{\mathbf{504}.}\]

\[x = \sqrt[3]{- \frac{b}{2} + \sqrt{\frac{b^{2}}{4} + \frac{a^{3}}{27}}} +\]

\[+ \sqrt[3]{- \frac{b}{2} - \sqrt{\frac{b^{2}}{4} + \frac{a^{3}}{27}}}\]

\[Пусть\ y = \sqrt[3]{- \frac{b}{2} + \sqrt{\frac{b^{2}}{4} + \frac{a^{3}}{27}}};\ \]

\[\ z = \sqrt[3]{- \frac{b}{2} - \sqrt{\frac{b^{2}}{4} + \frac{a^{3}}{27}}};\]

\[y^{3} + z^{3} = - \frac{b}{2} + \sqrt{\frac{b^{2}}{4} + \frac{a^{3}}{27}} - \frac{b}{2} -\]

\[- \sqrt{\frac{b^{2}}{4} + \frac{a^{3}}{27}} = - b.\]

\[x^{3} = (y + z)^{3} = y^{3} + 3y^{2}z +\]

\[+ 3yz^{2} + z^{3} = y^{3} + z^{3} +\]

\[+ 3yz(y + z) =\]

\[= - b + 3 \cdot \left( - \frac{a}{3} \right)x = - b - ax\]

\[x^{3} + ax + b = - b - ax +\]

\[+ ax + b = 0.\]

\[Ответ:0.\]