a) Ta có: \(\left(a-b\right)^3+\left(b-c\right)^3+\left(c-a\right)^3\)
\(=a^3-3a^2b+3ab^2-b^3+b^3-3b^2c+3bc^2-c^3+c^3-3c^2a+3ca^2-a^3\)
\(=3\left(a^2b+ac^2-ab^2-bc^2+b^2c-a^2c\right)\)
\(=3\left[\left(a^2b-ab^2\right)+\left(ac^2-bc^2\right)-\left(a^2c-b^2c\right)\right]\)
\(=3\left[ab\left(a-b\right)+c^2\left(a-b\right)-c\left(a^2-b^2\right)\right]\)
\(=3\left[ab\left(a-b\right)+c^2\left(a-b\right)-c\left(a-b\right)\left(a+b\right)\right]\)
\(=3\left(a-b\right)\left[ab+c^2-c\left(a+b\right)\right]\)
\(=3\left(a-b\right)\left(ab+c^2-ca-cb\right)\)
\(=3\left(a-b\right)\left[\left(ab-ac\right)+\left(c^2-cb\right)\right]\)
\(=3\left(a-b\right)\left[a\left(b-c\right)+c\left(c-b\right)\right]\)
\(=3\left(a-b\right)\left[a\left(b-c\right)-c\left(b-c\right)\right]\)
\(=3\left(a-b\right)\left(b-c\right)\left(a-c\right)\)
e) Ta có: \(\left(x+y+z\right)^3-x^3-y^3-z^3\)
\(=x^3+y^3+z^3+3\left(x+y\right)+3\left(y+z\right)+3\left(z+x\right)-x^3-y^3-z^3\)
\(=3\left(x+y+y+z+z+x\right)\)
\(=3\left(2x+2y+2z\right)\)
\(=6\left(x+y+z\right)\)
đ. \(\left(a-b\right)^3+\left(b-c\right)^3+\left(c-a\right)^3\\ =a^3-3a^2b+3ab^2-b^3+b^3-3b^2c+3bc^2-c^3+c^3-3c^2a+3ca^2-a^3\\=a^3-a^3+b^3-b^3+c^3-c^3-3a^2b+3ab^2-3b^2c+3bc^2-3c^2a+3ca^2\\ =-3\left(a^2b-ab^2+b^2c-bc^2+c^2a-ca^2\right)\\ =-3\left[ab\left(a-b\right)+c^2a-bc^2-\left(ca^2-b^2c\right)\right]\\ =-3\left[ab\left(a-b\right)+c^2\left(a-b\right)-c\left(a-b\right)\left(a+b\right)\right]\\ =-3\left(a-b\right)\left(ab+c^2-ac-bc\right)\\ =-3\left(a-b\right)\left[a\left(b-c\right)-c\left(b-c\right)\right]\\ =-3\left(a-b\right)\left(b-c\right)\left(a-c\right)\)
