Toán Lớp 8: rút gọn: (b-c)^3+(c-a)^3-(a-b)^3-3(a-b)(b-c)(c-a)
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Toán Lớp 8: rút gọn: (b-c)^3+(c-a)^3-(a-b)^3-3(a-b)(b-c)(c-a)
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{\left( {b – c} \right)^3} + {\left( {c – a} \right)^3} – {\left( {a – b} \right)^3}\\
– 3\left( {a – b} \right)\left( {b – c} \right)\left( {c – a} \right)\\
= {b^3} – 3{b^2}c + 3b{c^2} – {c^3} + {c^3} – 3a{c^2} + 3{a^2}c – {a^3}\\
– {a^3} + 3{a^2}b – 3a{b^2} + {b^3} – 3\left( {a – b} \right)\left( {bc – ab + ac – {c^2}} \right)\\
= 2{b^3} + 3{a^2}b – 3a{b^2} – 3{b^2}c + 3b{c^2} – 3a{c^2} + 3{a^2}c\\
– 3\left( {abc – {a^2}b + {a^2}c – a{c^2} – {b^2}c + a{b^2} – abc + b{c^2}} \right)\\
= 2{b^3} + 3{a^2}b – 3a{b^2} – 3{b^2}c + 3b{c^2} – 3a{c^2} + 3{a^2}c\\
+ 3{a^2}b – 3a{b^2} + 3{b^2}c – 3b{c^2} – 3{a^2}c + 3a{c^2}\\
= 2{b^3} + 6{a^2}b – 6a{b^2}
\end{array}$