Question

phân tích thành nhân tử

(x-y)\(^3\)+(y-z)\(^3\)+(z-x)\(^3\)

Answer 1

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

=> a+b+c=0 \(\Rightarrow a+b=-c\Rightarrow\left(a+b\right)^3=-c^3\)

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

\(\Rightarrow a^3+b^3+3ab\left(a+b\right)+c^3=0\)

\(\Rightarrow a^3+b^3+c^3+3ab\left(-c\right)=0\Rightarrow a^3+b^3+c^3-3abc=0\)

\(\Rightarrow a^3+b^3+c^3=3abc\)

\(\Leftrightarrow\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3=3\left(x-y\right)\left(y-z\right)\left(z-x\right)\)