Question

Đã trả lời bởi chuyên gia

Answer 1

\[\begin{array}{l}
do:a + b + c = 0\\
= > a + b = - c\\
= > {\left( {a + b} \right)^5} = - {c^5}\\
\Leftrightarrow {a^5} + 5{a^4}b + 10{a^3}{b^2} + 10{a^2}{b^3} + 5a{b^4} + {b^5} = - {c^5}\\
\Leftrightarrow {a^5} + {b^5} + {c^5} = - 5ab\left( {{a^3} + 2{a^2}b + 2a{b^2} + {b^3}} \right)\\
\Leftrightarrow {a^5} + {b^5} + {c^5} = - 5ab\left[ {{a^2}\left( {a + b} \right) + ab\left( {a + b} \right) + {b^2}\left( {a + b} \right)} \right]\\
\Leftrightarrow {a^5} + {b^5} + {c^5} = - 5ab\left[ {{a^2}\left( { - c} \right) + ab\left( { - c} \right) + {b^2}\left( { - c} \right)} \right]\\
\Leftrightarrow {a^5} + {b^5} + {c^5} = 5abc\left( {{a^2} + ab + {b^2}} \right)\\
\Leftrightarrow 2\left( {{a^5} + {b^5} + {c^5}} \right) = 5abc.2\left( {{a^2} + ab + {b^2}} \right)\\
\Leftrightarrow 2\left( {{a^5} + {b^5} + {c^5}} \right) = 5abc\left( {{a^2} + {b^2} + {a^2} + {b^2} + 2ab} \right)\\
\Leftrightarrow 2\left( {{a^5} + {b^5} + {c^5}} \right) = 5abc\left[ {{a^2} + {b^2} + {{\left( {a + b} \right)}^2}} \right]\\
\Leftrightarrow 2\left( {{a^5} + {b^5} + {c^5}} \right) = 5abc\left[ {{a^2} + {b^2} + {{\left( { - c} \right)}^2}} \right]\\
\Leftrightarrow 2\left( {{a^5} + {b^5} + {c^5}} \right) = 5abc\left( {{a^2} + {b^2} + {c^2}} \right)\\
Vay = > dpcm
\end{array}\]

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Answer 2

\[\begin{array}{l}
do:a + b + c = 0\\
= > a + b = - c\\
= > {\left( {a + b} \right)^5} = - {c^5}\\
\Leftrightarrow {a^5} + 5{a^4}b + 10{a^3}{b^2} + 10{a^2}{b^3} + 5a{b^4} + {b^5} = - {c^5}\\
\Leftrightarrow {a^5} + {b^5} + {c^5} = - 5ab\left( {{a^3} + 2{a^2}b + 2a{b^2} + {b^3}} \right)\\
\Leftrightarrow {a^5} + {b^5} + {c^5} = - 5ab\left[ {{a^2}\left( {a + b} \right) + ab\left( {a + b} \right) + {b^2}\left( {a + b} \right)} \right]\\
\Leftrightarrow {a^5} + {b^5} + {c^5} = - 5ab\left[ {{a^2}\left( { - c} \right) + ab\left( { - c} \right) + {b^2}\left( { - c} \right)} \right]\\
\Leftrightarrow {a^5} + {b^5} + {c^5} = 5abc\left( {{a^2} + ab + {b^2}} \right)\\
\Leftrightarrow 2\left( {{a^5} + {b^5} + {c^5}} \right) = 5abc.2\left( {{a^2} + ab + {b^2}} \right)\\
\Leftrightarrow 2\left( {{a^5} + {b^5} + {c^5}} \right) = 5abc\left( {{a^2} + {b^2} + {a^2} + {b^2} + 2ab} \right)\\
\Leftrightarrow 2\left( {{a^5} + {b^5} + {c^5}} \right) = 5abc\left[ {{a^2} + {b^2} + {{\left( {a + b} \right)}^2}} \right]\\
\Leftrightarrow 2\left( {{a^5} + {b^5} + {c^5}} \right) = 5abc\left[ {{a^2} + {b^2} + {{\left( { - c} \right)}^2}} \right]\\
\Leftrightarrow 2\left( {{a^5} + {b^5} + {c^5}} \right) = 5abc\left( {{a^2} + {b^2} + {c^2}} \right)\\
Vay = > dpcm
\end{array}\]

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