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20 tháng 10 2018

   

       \(xyz-\left(xy+yz+xz\right)+\left(x+y+z\right)-1\)

\(=\left(xyz-xy-xz+x\right)-yz+y+z-1\)

\(=x\left(yz-y-z+1\right)-\left(yz-y-z+1\right)\)

\(=\left(x-1\right)\left(yz-y-z+1\right)\)

\(=\left(x-1\right)\left[y\left(z-1\right)-\left(z-1\right)\right]\)

\(=\left(x-1\right)\left(y-1\right)\left(z-1\right)\)

27 tháng 6 2016

xy(x+y)+yz(y+z)+xz(x+z)+2xyz
= xy(x + y) + yz(y + z) + xyz + xz(x + z) + xyz
= xy(x + y) + yz(y + z + x) + xz(x + z + y)
= xy(x + y) + z(x + y + z)(y + x)
= (x + y)(xy + zx + zy + z2)
= (x + y)[x(y + z) + z(y + z)]
= (x + y)(y + z)(z + x)

27 tháng 6 2016

\(xy.\left(x+y\right)+yz.\left(y+z\right)+xz.\left(x+z\right)+2xyz\)
\(\Leftrightarrow x^2y+xy^2+y^2z+yz^2+x^2z+xz^2+2xyz\)

\(\Leftrightarrow xy\left(x+y\right)+xyz+yz\left(y+z\right)+xyz+xz\left(z+x\right)\)

\(\Leftrightarrow xy\left(x+y+z\right)+yz\left(x+y+z\right)+xz\left(x+z\right)\)

\(\Leftrightarrow y\left(x+y+z\right)\left(x+z\right)+xz\left(x+z\right)\)
\(\Leftrightarrow\left(x+z\right)\left(y\left(z+x\right)+zx\right)\)

\(\Leftrightarrow\left(x+z\right)\left(y+z\right)\left(x+y\right)\)

27 tháng 6 2016

\(xy\left(x+y\right)+yz\left(y+z\right)+xz\left(x+z\right)+2xyz\)

\(=xy.x+xy.y+yz.y+yz.z+xz.x+xz.z+2xyz\)

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

9 tháng 10 2019

Đặt x^2+y^2+z^2 =a ; xy+yz+zx=b

=> (x+y+z)^2 =x^2+y^2+z^2+2xy+2yz+2zx =a+2b

Ta có A= (x^2+y^2+z^2)(xy+yz+zx) +(x+y+z)^2

= a(a+2b)+b^2=a^2+2ab+b^2=(a+b)^2

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

16 tháng 12 2018

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

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

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

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

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

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

2 tháng 7 2021

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)