Question

Phân tích đa thức thành nhân tử 

\(\left(x-y\right)^5+\left(y-z\right)^5+\left(z-x\right)^5\)

Answer 1

Đặt : \(x-y=a\)\(,y-z=b\)

\(\Rightarrow z-x=-\left(a+b\right)\)

\(\left(x-y\right)^5+\left(y-z\right)^5+\left(z-x\right)^5=a^5+b^5\left[-\left(a+b\right)\right]^5=a^5+b^5-\left(a+b\right)^5\)

\(=a^5+b^5-\left(a^5+5a^4\times b+10a^3\times b^2+10a^2\times b^3+5a\times b^4+b^5\right)\)

\(=-\left(5a^4\times b+10a^3\times b^2+10a^2\times b^3+5a\times b^4\right)\)

\(=-5ab\left(a^3+2a^2\times b+2a\times b^2+b^3\right)\)

\(=-5ab\left[\left(a+b\right)\times\left(a^2+b^2-ab\right)+2ab\times\left(a+b\right)\right]\)

\(=-5ab\times\left(a+b\right)\times\left(a^2+ab+b^2\right)\)