Câu hỏi của nguyễn khánh linh - Toán lớp 8 - Học toán với OnlineMath

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

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

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

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

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) 

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

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

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

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

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

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

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

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

Tham khảo nhé~