Chứng minh a^3 + b^3 + c^3 = 3abc
Chứng minh \({a^3} + {b^3} + {c^3} = 3abc\).
Ta có:
\({a^3} + {b^3} + {c^3} - 3abc = \left( {{a^3} + {b^3}} \right) + {c^3} - 3abc\)
\( = {\left( {a + b} \right)^3} - 3ab\left( {a + b} \right) + {c^3} - 3abc = \left[ {{{\left( {a + b} \right)}^3} + {c^3}} \right] - \left[ {3ab\left( {a + b} \right) - 3abc} \right]\)
\( = \left( {a + b + c} \right)\left[ {{{\left( {a + b} \right)}^2} - \left( {a + b} \right)c + {c^2}} \right] - 3ab\left( {a + b + c} \right)\)
\( = \left( {a + b + c} \right)\left( {{a^2} + {b^2} + {c^2} - ab - bc - ca} \right)\)
Mà a + b + c = 0
\( \Rightarrow {a^3} + {b^3} + {c^3} - 3abc = 0\)
\( \Rightarrow {a^3} + {b^3} + {c^3} = 3abc\) (đpcm).