Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
x = \sqrt[3]{{5 - \sqrt {17} }} + \sqrt[3]{{5 + \sqrt {17} }}\\
\Leftrightarrow {x^3} = {\left( {\sqrt[3]{{5 - \sqrt {17} }} + \sqrt[3]{{5 + \sqrt {17} }}} \right)^3}\\
\Leftrightarrow {x^3} = {\sqrt[3]{{5 - \sqrt {17} }}^3} + 3.{\sqrt[3]{{5 - \sqrt {17} }}^2}.\sqrt[3]{{5 + \sqrt {17} }} + 3.\sqrt[3]{{5 - \sqrt {17} }}.{\sqrt[3]{{5 + \sqrt {17} }}^2} + {\sqrt[3]{{5 - \sqrt {17} }}^3}\\
\Leftrightarrow {x^3} = \left( {5 - \sqrt {17} } \right) + 3.\sqrt[3]{{5 - \sqrt {17} }}.\sqrt[3]{{5 + \sqrt {17} }}.\left( {\sqrt[3]{{5 - \sqrt {17} }} + \sqrt[3]{{5 + \sqrt {17} }}} \right) + \left( {5 + \sqrt {17} } \right)\\
\Leftrightarrow {x^3} = 5 - \sqrt {17} + 3.\sqrt[3]{{\left( {5 - \sqrt {17} } \right)\left( {5 + \sqrt {17} } \right)}}.x + 5 + \sqrt {17} \\
\Leftrightarrow {x^3} = 10 + 3.\sqrt[3]{{{5^2} - {{\sqrt {17} }^2}}}.x\\
\Leftrightarrow {x^3} = 10 + 3.\sqrt[3]{8}.x\\
\Leftrightarrow {x^3} = 10 + 3.2.x\\
\Leftrightarrow {x^3} - 6x - 10 = 0
\end{array}\)
Vậy \({x^3} - 6x - 10 = 0\)
