Đặt : \(x-y=a\)\(,y-z=b\)

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

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

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

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

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

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

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

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

Xét phương trình: \(\left(x+y+z\right)^5-x^5-y^5-z^5=0\)

Có nghiệm: \(x=-y;x=-z;y=-z\)

Hệ số của mũ là: 5

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

\(=5\left(x+y\right)\left(y+z\right)\left(z+x\right)\left(x^2+y^2+z^2+xy+yz+xz\right)\)

Hok Tốt!!!

để lâu cứt trâu hoá bùn

Thằng ngáo lol

a)  \(\left(x+y\right)^5-x-y=\left(x+y\right)^5-\left(x+y\right)=\left(x+y\right)\left[\left(x+y\right)^4-1\right]\)

= \(\left(x+y\right)\left(x+y-1\right)\left(x+y+1\right)\)     #áp dụng hàng đẳng thức#

c) \(x^9-x^7-x^6-x^5+x^4+x^3+x^2+1\)nhóm vào là đc

b) \(\left(x^2+y^2\right)^3+\left(z^2-x^2\right)^3+\left(y^2+z^2\right)^3\)

=\(\left(y^2+x^2\right)\left[\left(x^2+y^2\right)^2-\left(x^2+y^2\right)\left(z^2-x^2\right)+\left(z^2-x^2\right)^2\right]+\left(y^2+z^2\right)^3\)

= \(\left(y^2+z^2\right)\left[x^4+y^4+2x^2y^2-x^2z^2+x^4-y^2z^2+x^2y^2+z^4+x^4-2x^2z^2+y^4+z^4+2y^2z^2\right]\)

=\(=\left(y^2+z^2\right)\left(2x^4+2y^4+2z^4+3x^2y^2-3x^2z^2+y^2z^2\right)\)

a) \(x^7+x^5+x^4+x^3+x^2+1\)

\(=\left(x^7+x^4\right)+\left(x^5+x^2\right)+\left(x^3+1\right)\)

\(=x^4\left(x^3+1\right)+x^2\left(x^3+1\right)+\left(x^3+1\right)\)

\(=\left(x^3+1\right)\left(x^4+x^2+1\right)\)

\(=\left(x+1\right)\left(x^2-x+1\right)\left(x^4+x^2+1\right)\)