Question

Phân tích đa thức thành nhân tử:

A = (x+y+z)³-(x+y-z)³-(x-y+z)³-(-x+y+z)^3.

Answer 1

Đặt \(x+y-z=a;x-y+z=b;-x+y+z=c\) thì a + b + c = x + y + z

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

\(=\left(a+b+c-a\right)\left[\left(a+b+c\right)^2+a\left(a+b+c\right)+a^2\right]-\left(b^3+c^3\right)\)

\(=\left(b+c\right)\left[a^2+b^2+c^2+2\left(ab+bc+ca\right)+\left(a^2+ab+ac\right)+a^2\right]-\left(b+c\right)\left(b^2-bc+c^2\right)\)\(=\left(b+c\right)\left[3a^2+b^2+c^2+3ab+2bc+3ac-b^2+bc-c^2\right]\)

\(=\left(b+c\right)\left(3a^2+3ab+3bc+3ca\right)\)

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

Vậy..