Biểu thức \({(a + b + c)^3}\)được phân tích thành
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A.
\({a^3} + {b^3} + {c^3} + 3(a + b + c)\).
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B.
\({a^3} + {b^3} + {c^3} + 3(a + b)(b + c)(c + a)\).
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C.
\({a^3} + {b^3} + {c^3} + 6(a + b + c)\).
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D.
\({a^3} + {b^3} + {c^3} + 3({a^2} + {b^2} + {c^2}) + 3\left( {a + b + c} \right)\).
\(\begin{array}{c}{(a + b + c)^3} = {{\rm{[}}(a + b) + c{\rm{]}}^3}\\ = {(a + b)^3} + 3{(a + b)^2}c + 3(a + b){c^2} + {c^3}\\ = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} + 3{(a + b)^2}c + 3(a + b){c^2} + {c^3}\\ = {a^3} + {b^3} + {c^3} + 3ab(a + b) + 3{(a + b)^2}c + 3(a + b){c^2}\\ = {a^3} + {b^3} + {c^3} + 3(a + b)\left[ {ab + (a + b)c + {c^2}} \right]\\ = {a^3} + {b^3} + {c^3} + 3(a + b)(ab + ac + bc + {c^2})\\ = {a^3} + {b^3} + {c^3} + 3(a + b)\left[ {a(b + c) + c(b + c)} \right]\\ = {a^3} + {b^3} + {c^3} + 3(a + b)(b + c)(c + a)\end{array}\)
Vậy \({(a + b + c)^3}\) = \({a^3} + {b^3} + {c^3} + 3(a + b)(b + c)(c + a)\)
Đáp án : B