phân tích thành nhân tử x^2(y+z)+y^2(z+x)+z^2(x+y)+2xyz=0
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S
0
DT
1
2 tháng 8 2021
\(x^2y+y^2x+x^2z+z^2x+y^2z+z^2y+2xyz\)..
\(=\left(x^2y+z^2y+2xyz\right)+\left(y^2x+y^2z\right)+\left(z^2x+x^2z\right)\).
\(=y\left(x+z\right)^2+y^2\left(x+z\right)+xz\left(x+z\right)\)
\(=\left(xy+yz\right)\left(x+z\right)+\left(x+z\right)\left(y^2+xz\right)\).
\(=\left(x+z\right)\left(xy+yz+y^2+xz\right)\).
\(=\left(x+z\right)\left[x\left(y+z\right)+y\left(y+z\right)\right]\).
\(=\left(x+z\right)\left(x+y\right)\left(y+z\right)\).
PT
1
OP
9 tháng 8 2016
\(x^2y+xy^2+x^2z+y^2z+2xyz=z\left(x^2+2xy+y^2\right)+xy\left(x+y\right)=z\left(x+y\right)^2+xy\left(x+y\right)=\left(x+y\right)\left[z\left(x+y\right)+xy\right]=\left(x+y\right)\left(zx+zy+xy\right)\)
NM
0
PT
1
CM
7 tháng 10 2018
xy(x + y) + yz(y + z) + xz(x + z) + 2xyz
= x 2 y + x y 2 + yz(y + z) + x 2 z + x z 2 + xyz + xyz
= ( x 2 y + x 2 z) + yz(y + z) + (x y 2 + xyz) + (x z 2 + xyz)
= x 2 (y + z) + yz(y + z) + xy(y+ z) + xz(y + z)
= (y + z)( x 2 + yz + xy + xz) = (y + z)[( x 2 + xy) + (xz + yz)]
= (y + z)[x(x + y) + z(x + y)] = (y + z)(x+ y)(x + z)
N
0
TA
28 tháng 7 2017
1 ) \(x^2-x-y^2-y=\left(x^2-y^2\right)+\left(-x-y\right)=\left(x+y\right)\left(x-y\right)-\left(x+y\right)=\left(x+y\right)\left(x-y-1\right)\)
2 ) \(x^2-2xy+y^2-z^2=\left(x-y\right)^2-z^2=\left(x-y+z\right)\left(x-y-z\right)\)
3 ) \(5x-5y+ax-ay=5.\left(x-y\right)+a\left(x-y\right)=\left(x-y\right)\left(5+a\right)\)
4 ) \(a^3-a^2x-ay+xy=a^2.\left(a-x\right)-y.\left(a-x\right)=\left(a-x\right)\left(a^2-y\right)\)
5 ) \(xy.\left(x+y\right)+yz.\left(y+z\right)+xz.\left(x+z\right)+2xyz\)
\(=xy.\left(x+y\right)+y^2z+yz^2+x^2z+xz^2+xyz+xyz\)
\(=xy.\left(x+y\right)+\left(y^2z+xyz\right)+\left(yz^2+xz^2\right)+\left(x^2z+xyz\right)\)
\(=xy.\left(x+y\right)+yz.\left(x+y\right)+z^2.\left(x+y\right)+xz.\left(x+y\right)\)
\(=\left(x+y\right)\left(xy+yz+z^2+xz\right)=\left(x+y\right)\left[\left(xy+xz\right)+\left(yz+z^2\right)\right]\)
\(=\left(x+y\right)\left[x.\left(y+z\right)+z.\left(y+z\right)\right]=\left(x+y\right)\left(y+z\right)\left(x+z\right)\)
NH
7 tháng 10 2019
a, \(xy\left(x+y\right)+yz\left(y+z\right)+xz\left(x+z\right)+2xyz\)\(=x^2y+xy^2+y^2z+yz^2+x^2z+xz^2+2xyz\)
\(=\left(x^2y+xy^2+xyz\right)+\left(x^2z+xz^2+xyz\right)+\left(y^2z+yz^2\right)\)
\(=xy\left(x+y+z\right)+xz\left(x+z+y\right)+yz\left(y+z\right)\)
\(=x\left(x+y+z\right)\left(y+z\right)+yz\left(y+z\right)\)
\(=\left(y+z\right)\left(x^2+xy+xz+yz\right)\)
\(=\left(y+z\right)\left[x\left(x+z\right)+y\left(x+z\right)\right]\)
\(=\left(y+z\right)\left(x+z\right)\left(x+y\right)\)
b, \(2x^2+2y^2-x^2z+z-y^2z-2\)
\(=\left(2x^2-x^2z\right)+\left(2y^2-y^2z\right)-\left(2-z\right)\)
\(=x^2\left(2-z\right)+y^2\left(2-z\right)-\left(2-z\right)\)
\(=\left(2-z\right)\left(x^2+y^2-1\right)\)
x^2y+xy^2+x^2z+xz^2+y^2z+yz^2+2xyz
=x^2y+xy^2+xyz+x^2z+xz^2+xyz+y^2z+yz^2
=xy(x+y+z)+zx(x+y+z)+yz(y+z)
=x(y+z)(x+y+z)+yz(y+z)
=(y+z)(x^2+xy+zx+yz)
=(x+y)(y+z)(z+x)