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\(=a\left(b+c\right)\left(b^2-c^2\right)+b\left(c+a\right)\left(c^2-a^2\right)+c\left(a+b\right)\left(a^2-b^2\right)\)
\(=\left(ab+ac\right)\left(b^2-c^2\right)+\left(bc+ba\right)\left(c^2-a^2\right)+\left(ca+cb\right)\left(a^2-b^2\right)\)
\(=ab^3+ab^2c-abc^2-ac^3+bc^3+abc^2-a^2bc-a^3b+a^3c+a^2bc-ab^2c-b^3c\)
\(=ab^3-ac^3+bc^3-a^3b+a^3c-b^3c\)
\(=\left(ab^3-b^3c\right)+\left(bc^3-ac^3\right)+\left(a^3c-a^3b\right)\)
\(=b^3\left(a-c\right)+c^3\left(b-a\right)+a^3\left(c-b\right)\)
\(=b^3\left(a-c\right)+c^3\left(c-a+b-c\right)+a^3\left(c-b\right)\)(Đổi dấu hạng tử ở giữa)
\(=b^3\left(a-c\right)-c^3\left(a-c\right)-c^3\left(c-b\right)+a^3\left(c-b\right)\)
\(=\left(a-c\right)\left(b^3-c^3\right)-\left(b-c\right)\left(a^3-c^3\right)\)
\(=\left(a-c\right)\left(b-c\right)\left(b^2+bc+c^2\right)-\left(a-c\right)\left(b-c\right)\left(a^2+ac+c^2\right)\)
\(=\left(a-c\right)\left(b-c\right)\left(b^2+bc+c^2-a^2-ac-c^2\right)\)
\(=\left(a-c\right)\left(b-c\right)\left(b^2-a^2-ac+bc\right)\)
\(=\left(a-c\right)\left(b-c\right)[\left(b-a\right)\left(b+a\right)+c\left(b-a\right)]\)
\(=\left(a-c\right)\left(b-c\right)\left(b-a\right)\left(a+b+c\right)\)
\(\left(a+b\right)\left(a^2-b^2\right)+\left(b+c\right)\left(b^2-c^2\right)+\left(c+a\right)\left(c^2-a^2\right)\)
\(=\left(a+b\right)\left(a^2-b^2\right)-\left(b+c\right)\left[c^2-a^2+a^2-b^2\right]+\left(c+a\right)\left(c^2-a^2\right)\)
\(=\left(a+b\right)\left(a^2-b^2\right)-\left(b+c\right)\left(c^2-a^2\right)-\left(b+c\right)\left(a^2-b^2\right)+\left(c+a\right)\left(c^2-a^2\right)\)
\(=\left(a^2-b^2\right)\left(a+b-b-c\right)+\left(c^2-a^2\right)\left(c+a-b-c\right)\)
\(=\left(a-b\right)\left(a+b\right)\left(a-c\right)+\left(c-a\right)\left(c+a\right)\left(a-b\right)\)
\(=\left(a-b\right)\left(a-c\right)\left(a+b-c-a\right)\)
\(=\left(a-b\right)\left(a-c\right)\left(b-c\right)\)
Chúc bạn học tốt.
1) \(\left(a-b\right)\left(c-a\right)\left(c-b\right)\left(c+b+a\right)\)