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\(\left(a-b\right)\left(b-c\right)\left(a-c\right)+\left(a+b\right)\left(b+c\right)\left(a-c\right)+\left(c+a\right)\left(a+b\right)\left(b-c\right)\)
\(=\left(a-c\right)\left[\left(a-b\right)\left(b-c\right)+\left(a+b\right)\left(b+c\right)\right]+\left(c+a\right)\left(a+b\right)\left(b-c\right)\)
\(=\left(a-c\right)\left(2ab+2bc\right)+\left(c+a\right)\left(a+b\right)\left(b-c\right)\)
\(=2b\left(a-c\right)\left(a+c\right)+\left(c+a\right)\left(a+b\right)\left(b-c\right)\)
\(=\left(a+c\right)\left[2b\left(a-c\right)+\left(a+b\right)\left(b-c\right)\right]\)
Ta có: \(\left(a-b\right)\left(b-c\right)\left(a-c\right)+\left(a+b\right)\left(b+c\right)\left(a-c\right)+\left(a+b\right)\left(a+c\right)\left(c-b\right)\)
\(=\left(a-c\right).\left[\left(a-b\right)\left(b-c\right)+\left(a+b\right)\left(b+c\right)\right]+\left(a+b\right)\left(a+c\right)\left(c-b\right)\)
\(=\left(a-c\right).\left(ab-ac-b^2+bc+ab+ac+b^2+bc\right)+\left(a+b\right)\left(a+c\right)\left(c-b\right)\)
\(=\left(a-c\right).\left(2ab+2bc\right)+\left(a+b\right)\left(a+c\right)\left(c-b\right)\)
\(=2b.\left(a-c\right).\left(a+c\right)+\left(a+b\right)\left(a+c\right)\left(c-b\right)\)
\(=\left(a+c\right)\left[2b\left(a-c\right)+\left(a+b\right)\left(c-b\right)\right]\)
\(=\left(a+c\right)\left(2ab-2bc+ac-ab+bc-b^2\right)\)
\(=\left(a+c\right)\left(ab-bc+ac-b^2\right)\)
\(=\left(a+c\right)\left[a.\left(b+c\right)-b.\left(b+c\right)\right]\)
\(=\left(a+c\right)\left(a-b\right)\left(b+c\right)\)
(a-b)(b-c)(a-c)+(a+b)(c+a)(c-b)+(b+c)(c+a)(b-a)
= ( b - c) [ a2−ab−ac+bc−ac−bc−a2−ab]+(b+c)(c+a)(b−a)a2−ab−ac+bc−ac−bc−a2−ab]+(b+c)(c+a)(b−a)
= -2a.(b + c) (b - c) + (b+c)(c+a)(b-a)
= ( b + c ) ( bc + ab - ac - a2a2 - 2ab + 2ac )
= ( b + c ) ( bc - ab + ac - a2a2 )
= ( b + c ) ( a + b ) ( c - a )
minhf chắc sev d