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

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

\(=-xy^2+x^2y-yz^2+y^2z-zx^2+z^2x\)

\(=\left(x-y\right)\left(z-x\right)\left(z-y\right)\)

nâng cao phát triển toán 8 tập 1 mình ngại viết nên bạn vào đó xem nhé

Ây za,mik ko bt có đúng ko nhưng mik thử làm nhé.

Đặt \(x^4+y^4+z^4=a;x^2+y^2+z^2=b;x+y+z=c\)

\(\Rightarrow M=2a-b^2-2bc^2+c^4\)

\(M=2a-2b^2+b^2-2bc^2+c^4\)

\(M=2\left(a-b^2\right)+\left(b-c^2\right)^2\)

Mà:

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

\(b-c^2=-2\left(xy+yz+zx\right)\)

Khi đó:

\(M=-4\left(x^2y^2+y^2z^2+z^2x^2\right)+4\left(xy+yz+zx\right)^2\)

\(M=-4x^2y^2-4y^2z^2-4z^2x^2+4x^2y^2++4y^2z^2+4z^2x^2+4z^2x^2+8x^2yz+8xy^2z+8xyz^2\)

\(M=8xyz\left(x+y+z\right)\)

a) xy(x + y) + yz(y + z) + xz(z + x) + 3xyz

= xy(X + y + z)  + yz(x + y + z) + xz(X + y + z)

= (x + y +z)(xy + yz+ xz)

b) xy(x + y) - yz(y + z) - xz(z - x)

= x²y + xy² - y²z - yz² - xz² + x²z

= x²(y + z) - yz(y + z) + x(y² - z²)

= x²(y + z) - yz(y + z) + x(y + z)(y - z)

= (y + z)(x² - yz + xy - xz)

= (y + z)[x(x + y) - z(x + y)]

= (y + z)(x + y)(x - z)

c) x(y² - z²) + y(z² - x²) + z(x² - y²)

 = x(y - z)(y + z) + yz² - yx² + x²z - y²z

= x(y - z)(y + z) - yz(y - z) - x²(y - z)

= (y - z)((xy + xz - yz - x²)

= (y - z)[x(y - x) - z(y - x)]

= (y - z)(x - z)(y -x) 

Làm như vầy là sai hướng rồi.

Tham khảo :

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   = 3(y²+z²)(x⁴+x²y²-x²z²-y²z²)

   = 3(y²+z²)[x²(x²+y²)-z²(x²+y²)]

   = 3(y²+z²)(x²-z²)(x²+y²)

   = 3(y²+z²)(x-z)(x+z)(x²+y²)

b) \(\left(x+y\right)^3-x^3-y^3\)

\(=x^3+3x^2y+3xy^2+y^3-x^3-y^3\)

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

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

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

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

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

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

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

  = (x+y)[3xy+3xz+3yz+3z² ] 

  = 3(x+y)(xy+xz+yz+z²)

  = 3(x+y)[x(y+z)+z(y+z)]

  = 3(x+y)(x+z)(y+z)

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

\(=x^6+3x^4y^2+3x^4y^2+y^6+z^2+-x^2+-y^6+-3y^4z+-3y^2z^4+-z^6\)

\(=x^6+3x^4y^2+3x^2y^4+-3y^4z^4+-z^6+-x^2+z^2\)

b) \(\left(x+y\right)^3-x^3-y^3\)

\(=x^3+3x^2y+3xy^2+y^3+-x^3+-y^3\)

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

\(=3x^2y+3xy^2\)

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

\(=x^3+3x^2y+3x^2z+3xy^2+6xyz+3xz^2+y^3+3y^2z+3yz^2+z^2-x^3-y^3-z^3\)

\(=3x^2y+3x^2z+3xy^2+3xy^2+6xyz+3xz^2+3y^2z+3yz^2\)

P/s: Ko chắc

cái này = (x+y)(y+z)(z+x)

cái này mình học nhìn quen rồi còn bạn giải từ chỗ mình vừa viết ở trên rồi giải ngược lại nhé

x(y²+z²)+y(x²+z²)+z(x²+y²) +2xyz= xy²+yx²+zx²+z²x+zy²+z²y+2xyz=(xyz +x²y+y²z+xy²)+(xyz+xz²+x²z+z²y)=y(xz+x²+yz+xy)+z(xy+x²+yz+xy)=(x+y)(xz+x²+yz+xy)=(x+y)(x(x+z)+y(x+z))=(x+y)(y+z)(z+x)