Toán Chứng minh (a+b+c)^3-a^3-b^3-c^3 = 3(a+b)(b+c)(c+a) 10/09/2021 By Madelyn Chứng minh (a+b+c)^3-a^3-b^3-c^3 = 3(a+b)(b+c)(c+a)
\(\begin{array}{l} \,\,\,\,\,{\left( {a + b + c} \right)^3} – {a^3} – {b^3} – {c^3} = 3\left( {a + b} \right)\left( {b + c} \right)\left( {c + a} \right)\\ VT = {\left( {a + b} \right)^3} + {c^3} + 3\left( {a + b} \right)c\left( {a + b + c} \right) – {a^3} – {b^3} – {c^3}\\ VT = {\left( {a + b} \right)^3} + 3\left( {a + b} \right)c\left( {a + b + c} \right) – {a^3} – {b^3}\\ VT = {a^3} + {b^3} + 3ab\left( {a + b} \right) + 3\left( {a + b} \right)c\left( {a + b + c} \right) – {a^3} – {b^3}\\ VT = 3ab\left( {a + b} \right) + 3\left( {a + b} \right)c\left( {a + b + c} \right)\\ VT = 3\left( {a + b} \right)\left[ {ab + c\left( {a + b + c} \right)} \right]\\ VT = 3\left( {a + b} \right)\left( {ab + ac + bc + {c^2}} \right)\\ VT = 3\left( {a + b} \right)\left[ {a\left( {b + c} \right) + c\left( {b + c} \right)} \right]\\ VT = 3\left( {a + b} \right)\left( {b + c} \right)\left( {c + a} \right) \end{array}\) Trả lời
\(\begin{array}{l}
\,\,\,\,\,{\left( {a + b + c} \right)^3} – {a^3} – {b^3} – {c^3} = 3\left( {a + b} \right)\left( {b + c} \right)\left( {c + a} \right)\\
VT = {\left( {a + b} \right)^3} + {c^3} + 3\left( {a + b} \right)c\left( {a + b + c} \right) – {a^3} – {b^3} – {c^3}\\
VT = {\left( {a + b} \right)^3} + 3\left( {a + b} \right)c\left( {a + b + c} \right) – {a^3} – {b^3}\\
VT = {a^3} + {b^3} + 3ab\left( {a + b} \right) + 3\left( {a + b} \right)c\left( {a + b + c} \right) – {a^3} – {b^3}\\
VT = 3ab\left( {a + b} \right) + 3\left( {a + b} \right)c\left( {a + b + c} \right)\\
VT = 3\left( {a + b} \right)\left[ {ab + c\left( {a + b + c} \right)} \right]\\
VT = 3\left( {a + b} \right)\left( {ab + ac + bc + {c^2}} \right)\\
VT = 3\left( {a + b} \right)\left[ {a\left( {b + c} \right) + c\left( {b + c} \right)} \right]\\
VT = 3\left( {a + b} \right)\left( {b + c} \right)\left( {c + a} \right)
\end{array}\)