Question

Phân tích đa thức thành nhân tử: \(3abc+a^2.\left(a-b-c\right)+b^2.\left(b-a-c\right)+c^2.\left(c-a-b\right)-c.\left(b-c\right).\left(a-c\right)\)

Answer 1

Lời giải:

$3abc+a^2(a-b-c)+b^2(b-a-c)+c^2(c-a-b)-c(b-c)(a-c)$

$=3abc+(a^3+b^3+c^3)-[a^2(b+c)+b^2(c+a)+c^2(a+b)]-c(ab-bc-ac+c^2)$

$=2abc+a^3+b^3-[a^2(b+c)+b^2(c+a)]$

$=2abc+a^3+b^3-[ab(a+b)+c(a^2+b^2)]$

$=a^3+b^3-ab(a+b)+2abc-c(a^2+b^2)$

$=(a+b)(a^2-ab+b^2)-ab(a+b)-c(a^2+b^2-2ab)$

$=(a+b)(a^2-2ab+b^2)-c(a^2-2ab+b^2)$

$=(a+b)(a-b)^2-c(a-b)^2=(a-b)^2(a+b-c)$