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( x+y)(y+z)(z+x)+xyz
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NH
2
2 tháng 8 2021
\(z^3\left(x+y^2\right)+y^3\left(z-x^2\right)-x^3\left(y+z^2\right)-xyz\left(xyz-1\right)\)
\(=xz^3+y^2z^3+y^3z-x^2y^3-x^3-x^3z^2-x^2y^2z^2+xyz\)
\(=\left(y^2z^3+y^3z\right)+\left(xz^3+xyz\right)-\left(x^2y^3+x^2y^2z^2\right)-x^3\left(y+z^2\right)\)
\(=y^2z\left(y+z^2\right)+xz\left(y+z^2\right)-x^2y^2\left(y+z^2\right)-x^3\left(y+z^2\right)\)
\(=\left(y+z^2\right)\left(y^2z+xz-x^2y^2-x^3\right)\)
\(=\left(y+z^2\right)\left[z\left(y^2+x\right)-x^2\left(y^2+x\right)\right]\)
\(=\left(y+z^2\right)\left(z-x^2\right)\left(y^2+x\right)\)
Tick hộ nha bạn 😘
HT
0
NQ
1
MC
0
PV
4
28 tháng 6 2018
\(=\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\)
\(=\left(x+y\right)^3+z^3-3xy\left(x+y\right)-3xyz\)
\(=\left(x+y+z\right)\left[\left(x+y\right)^2+z\left(x+y\right)+z^2\right]-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)\left[\left(x+y\right)^2+z\left(x+y\right)+z^2-3xy\right]\)
\(=\left(x+y+z\right)\left(x^2+y^2+z^2+xz+yz-xy\right)\)
5 tháng 9 2018
\(x^3+y^3+z^3-3xyz\)
\(=\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\)
\(=\left(x+y+z\right)\left[\left(x+y\right)^2-\left(x+y\right)z+z^2\right]-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-zx\right)\)
10 tháng 12 2017
Ta có: \(\left(x+y\right)\left(y+z\right)\left(z+x\right)+xyz=x^2y+xy^2+xyz+y^2z+yz^2+xyz+xz^2+x^2x+xyz\)
\(=xy\left(x+y+z\right)+yz\left(x+y+z\right)+zx\left(x+y+z\right)=\left(x+y+z\right)\left(xy+yz+zx\right)\)
\(x^2-y^2+10x-6y+16=\left(x^2+10x+25\right)-\left(y^2+6y+9\right)\)
\(=\left(x+5\right)^2-\left(y+3\right)^2=\left(x+y+8\right)\left(x-y+2\right)\)
\(x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)=x^2\left(y-z\right)+yz\left(y-z\right)-x\left(y-z\right)\left(y+z\right)\)
\(=\left(y-z\right)\left(x^2+yz-xy-xz\right)=\left(y-z\right)\left(x-y\right)\left(z-x\right)\)
TQ
1
HN
22 tháng 8 2021
\(xyz-\left(xy+yz+xz\right)+\left(x+y+z\right)-1\)
\(=xyz-xy-yz+y-xz+x+z-1\)
\(=xy\left(z-1\right)-y\left(z-1\right)-x\left(z-1\right)+z-1\)
\(=\left(xy-y-x+1\right)\left(z-1\right)\)
\(=[\left(x-1\right)y-\left(x-1\right)]\left(z-1\right)\)
\(=\left(x-1\right)\left(y-1\right)\left(z-1\right)\)
QH
0
28 tháng 10 2015
y(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)
\(\left(x+y\right)\left(x+z\right)\left(y+z\right)+xyz\)
Khai triển ra ta được:
\(=\left[xyz+\left(xy^2+yx^2\right)+\left(xz^2+zx^2\right)+\left(yz^2+zy^2\right)+xyz\right]+xzy\)
\(=\left[xyz+xy\left(x+y\right)+xz\left(x+z\right)+yz\left(y+z\right)+xyz\right]+xyz+A+B\)
\(A=\left(xy+xz+yz\right)\)và \(B=\left(-xy-xz-yz\right)\)
\(=\left[xy\left(x+y\right)+xy\right]+\left[xz\left(x+z\right)+xz\right]+\left[yz\left(y+z\right)+yz\right]+\left(xyz-xy\right)+\left(xyz-xz\right)+\left(xyz-yz\right)\)
\(=xy\left(x+y+1\right)+xz\left(x+z+1\right)+yz\left(y+z+1\right)+xy\left(z-1\right)+xz\left(y-1\right)+yz\left(x-1\right)\)
\(=xy\left(x+y+z\right)+xz\left(x+z+y\right)+yz\left(y+z+x\right)\)
\(=\left(x+y+z\right)\left(xy+yz+zx\right)\)