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\(B=a\left(b^2-c^2\right)+b\left(c^2-a^2\right)+c\left(a^2-b^2\right)\)
\(B=ab^2-ac^2+bc^2-a^2b+a^2c-b^2c\)
\(B=\left(ab^2-a^2b\right)-\left(ac^2-c^2b\right)+\left(a^2c-b^2c\right)\)
\(B=-ab\left(a-b\right)-c^2\left(a-b\right)+c\left(a-b\right)\left(a+b\right)\)
\(B=\left(a-b\right)\left(-ab-c^2+ac+bc\right)\)
\(B=\left(a-b\right)\left[a\left(c-b\right)-c\left(c-b\right)\right]\)
\(B=\left(a-b\right)\left(c-b\right)\left(a-c\right)\)
\(a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)\)
\(=a^2\left(b-c\right)+b^2\left(c-a\right)-c^2\left[\left(b-c\right)+\left(c-a\right)\right]\)
\(=a^2\left(b-c\right)+b^2\left(c-a\right)-c^2\left(b-c\right)-c^2\left(c-a\right)\)
\(=\left(b-c\right)\left(a^2-c^2\right)+\left(c-a\right)\left(b^2-c^2\right)\)
\(=\left(b-c\right)\left(a-c\right)\left(a+c\right)+\left(c-a\right)\left(b-c\right)\left(b+c\right)\)
\(=\left(b-c\right)\left(a-c\right)\left(a+c-b-c\right)\)
\(=\left(a-b\right)\left(b-c\right)\left(a-c\right)\)
\(a\left(b^2-c^2\right)+b\left(c^2-a^2\right)+c\left(a^2-b^2\right)\)
\(=ab^2-ac^2+ca^2-cb^2+b\left(c^2-a^2\right)\)
\(=\left(ab^2-cb^2\right)-\left(ac^2-ca^2\right)+b\left(c-a\right)\left(c+a\right)\)
\(=b^2\left(a-c\right)-ac\left(c-a\right)+b\left(c-a\right)\left(c+a\right)\)
\(=b^2\left(a-c\right)+ac\left(a-c\right)-b\left(a-c\right)\left(c+a\right)\)
\(=\left(a-c\right)\left[b^2+ac-b\left(c+a\right)\right]\)
\(=\left(a-c\right)\left(b^2+ac-bc-ab\right)\)
\(=\left(a-c\right)\left[b\left(b-c\right)+a\left(c-b\right)\right]\)
\(=\left(a-c\right)\left[b\left(b-c\right)-a\left(b-c\right)\right]\)
\(=\left(a-c\right)\left(b-c\right)\left(b-a\right)\)
Cách khác:
Ta có:
\(a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)\)
\(=a(b^2-c^2)-b[(b^2-c^2)+(a^2-b^2)]+c(a^2-b^2)\)
\(=a(b^2-c^2)-b(b^2-c^2)-b(a^2-b^2)+c(a^2-b^2)\)
\(=(a-b)(b^2-c^2)-(b-c)(a^2-b^2)\)
\(=(a-b)(b-c)(b+c)-(b-c)(a-b)(a+b)\)
\(=(a-b)(b-c)[(b+c)-(a+b)]=(a-b)(b-c)(c-a)\)