Question

Phân tích đa thức thành nhân tử:

\(3abc+a^2\left(a-b-c\right)+b^2\left(b-a-c\right)+c2\left(c-b-a\right)-c\left(b-c\right)\left(a-c\right)\)

Answer 1

Đặt \(f=a^2\left(a-b-c\right)+b^2\left(b-a-c\right)+c^2\left(c-a-b\right)\)

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

\(=a^2\left(a-b\right)+b^2\left(b-a\right)+c\left[2ab-a^2-b^2+c\left(c^2-bc-ac+ab\right)\right]\)

\(=\left(a-b\right)\left(a^2-b^2\right)-c\left(a-b\right)^2+c\left(c-a\right)\left(c-b\right)\)

\(=\left(a-b\right)^2\left(a+b+c\right)+c\left(b-c\right)\left(a-c\right)\)

\(\Rightarrow BT=\left(a-b\right)^2\left(a+b+c\right)+c\left(b-c\right)\left(a-c\right)-c\left(b-c\right)\left(a-c\right)\)

\(=\left(a+b\right)^2\left(a+b+c\right)\)