Question

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

Answer 1

Để ngắn gọn, đặt \(\left\{{}\begin{matrix}b+c-a=x\\c+a-b=y\\a+b-c=z\end{matrix}\right.\) \(\Rightarrow x+y+z=a+b+c\)

Ta cần phân tích: \(\left(x+y+z\right)^3-x^3-y^3-z^3\)

\(=\left(y+z\right)\left[\left(x+y+z\right)^2+x\left(x+y+z\right)+x^2\right]-\left(y+z\right)\left(y^2-yz+z^2\right)\)

\(=\left(y+z\right)\left(3x^2+y^2+z^2+3xy+3xz+2yz-y^2+yz-z^2\right)\)

\(=\left(y+z\right)\left(3x^2+3xy+3yz+3zx\right)\)

\(=3\left(x+y\right)\left(y+z\right)\left(z+x\right)\)

\(=3\left(b+c-a+c+a-b\right)\left(c+a-b+a+b-c\right)\left(b+c-a+a+b-c\right)\)

\(=24abc\)