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\(yz\left(y+z\right)+zx\left(z-x\right)-xy\left(x+y\right)\)
\(=yz\left(y+z\right)+zx\left(z-x\right)-xy\left[\left(y+z\right)-\left(z-x\right)\right]\)
\(=yz\left(y+z\right)+zx\left(z-x\right)-xy\left(y+z\right)+xy\left(z-x\right)\)
\(=y\left(y+z\right)\left(z-x\right)+x\left(z-x\right)\left(z-y\right)\)
\(=\left(z-x\right)\left(yz-xy+xz-xy\right)\)
Đa thức trên tương đương với đa thức:
\(\left(xy\left(x+y\right)+xyz\right)+\left(yz\left(y+z\right)+xyz\right)+\left(xz\left(x+z\right)+xyz\right)\)
=\(xy\left(x+y+z\right)+yz\left(x+y+z\right)+xz\left(x+y+z\right)\)
=\(\left(x+y+z\right)\left(xy+yz+xz\right)\)
xy(x + y) + yz( y + z )+ zx( z + x ) + 3xyz
=xy(x + y) + xyz + yz(y + z) + xyz + xz(x + z)+xyz
=zy(x + y + z) + yz(x + y + z) + xz(x + y + z)
=(x + y + z)(xy + yz + zx)
chúc bn hok tốt
a) \(\left(x-y\right)^2+\left(y-z\right)^3+\left(z-x\right)^3\)
\(=\left(x-y\right)^2+\left(y-z+z-x\right)\left[\left(y-z\right)^2-\left(y-z\right)\left(z-x\right)+\left(z-x\right)^2\right]\)
\(=\left(x-y\right)^2+\left(y-x\right)\left(x^2+y^2+3z^2-3yz+xy-3xz\right)\)
\(=\left(x-y\right)\left(x-y-x^2-y^2-3z^2+3yz-xy+3xz\right)\)
Cô nghĩ phân tích đa thức này sẽ đẹp hơn:
\(\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3\)
\(=\left(x-y+y-z\right)\left[\left(x-y\right)^2-\left(x-y\right)\left(y-z\right)+\left(y-z\right)^2\right]+\left(z-x\right)^3\)
\(=\left(x-z\right)\left[\left(x-y\right)^2-\left(x-y\right)\left(y-z\right)+\left(y-z\right)^2\right]+\left(z-x\right)^3\)
\(=\left(x-z\right)\left[\left(x-y\right)^2-\left(x-y\right)\left(y-z\right)+\left(y-z\right)^2-\left(z-x\right)^2\right]\)
\(=\left(x-z\right)\left(3y^2-3xy+3zx-3xyz\right)\)
\(=3\left(x-y\right)\left(y-z\right)\left(z-x\right)\)
b) \(\left(x+y+z\right)\left(xy+yz+zx\right)-xyz\)
\(=\left(xy+yz+zx\right)\left(x+y+z\right)-xyz\)
\(=xy\left(x+y+z\right)+\left(yz+zx\right)\left(x+y+z\right)-xyz\)
\(=xy\left(x+y+z-z\right)+\left(yz+zx\right)\left(x+y+z\right)\)
\(=xy\left(x+y\right)+z\left(y+x\right)\left(x+y+z\right)\)
\(=\left(x+y\right)\left[xy+z\left(x+y+z\right)\right]\)
\(=\left(x+y\right)\left(xy+zx+zy+z^2\right)\)
\(=\left(x+y\right)\left[x\left(y+z\right)+z\left(y+z\right)\right]\)
\(=\left(x+y\right)\left(y+z\right)\left(z+x\right)\)
a) \left(x-y\right)^2+\left(y-z\right)^3+\left(z-x\right)^3(x−y)2+(y−z)3+(z−x)3
=\left(x-y\right)^2+\left(y-z+z-x\right)\left[\left(y-z\right)^2-\left(y-z\right)\left(z-x\right)+\left(z-x\right)^2\right]=(x−y)2+(y−z+z−x)[(y−z)2−(y−z)(z−x)+(z−x)2]
=\left(x-y\right)^2+\left(y-x\right)\left(x^2+y^2+3z^2-3yz+xy-3xz\right)=(x−y)2+(y−x)(x2+y2+3z2−3yz+xy−3xz)
=\left(x-y\right)\left(x-y-x^2-y^2-3z^2+3yz-xy+3xz\right)=(x−y)(x−y−x2−y2−3z2+3yz−xy+3xz
\left(x-y\right)^3+\left(y-z\right)^3+\left
=\left(x-y+y-z\right)\left[\left(x-y\right)^2-\left(x-y\right)\left(y-z\right)+\left(y-z\right)^2\right]+\left(z-x\right)^3
=\left(x-z\right)\left[\left(x-y\right)^2-\left(x-y\right)\left(y-z\right)+\left(y-z\right)^2\right]+\l
=\left(x-z\right)\left[\left(x-y\right)^2-\left(x-y\right)\left(y-z\right)+\left(y-z\right)^2-\left(z-x\
=\left(x-z\right)\left(
=3\left(x-y\right)\lefb) \left(x+y+z\right)\left(xy+yz+zx\right)-xyzb)(x+y+z)(xy+yz+zx)−xyz
=\left(xy+yz+zx\right)\left(x+y+z\right)-xyz=(xy+yz+zx)(x+y+z)−xyz
=xy\left(x+y+z\right)+\left(yz+zx\right)\left(x+y+z\right)-xyz=xy(x+y+z)+(yz+zx)(x+y+z)−xyz
=xy\left(x+y+z-z\right)+\left(yz+zx\right)\left(x+y+z\right)=xy(x+y+z−z)+(yz+zx)(x+y+z)
=xy\left(x+y\right)+z\left(y+x\right)\left(x+y+z\right)=xy(x+y)+z(y+x)(x+y+z)
=\left(x+y\right)\left[xy+z\left(x+y+z\right)\right]=(x+y)[xy+z(x+y+z)]
=\left(x+y\right)\left(xy+zx+zy+z^2\right)=(x+y)(xy+zx+zy+z2)
=\left(x+y\right)\left[x\left(y+z\right)+z\left(y+z\right)\right]=(x+y)[x(y+z)+z(y+z)]
=\left(x+y\right)\left(y+z\right)\left(z+x\right)=(x+y)(y+z)(z+x)
\(2xyz+x^2y+xy^2+x^2z+xz^2+y^2z+yz^2\)
\(=x^2\left(y+z\right)+yz\left(y+z\right)+x\left(y^2+z^3\right)+2xyz\)
\(=\left(y+z\right)\left(x^2+yz\right)+x\left(y^2+z^2+2yz\right)\)
\(=\left(y+z\right)\left(x^2+yz\right)+x\left(y+z\right)^2\)
\(=\left(y+z\right)\left(x^2+yz\right)+xy+xz\)
\(=\left(y+z\right)\left[x\left(x+2\right)+y\left(x+2\right)\right]\)
\(=\left(y+z\right)\left(x+y\right)\left(x+2\right)\)
\(b,x^2\left(y-z\right)+y^2\left(z-y\right)+z^2\left(x-y\right)\)
\(=x^2\left(y-z\right)+y^2z-y^2x+z^2x-z^2y\)
\(=x^2\left(y-z\right)+yz\left(y-z\right)-x\left(y^2-z^2\right)\)
\(=\left(y-z\right)\left[x^2+yz-x\left(y+z\right)\right]\)
\(=\left(y-z\right)\left[x\left(x-y\right)-z\left(x-y\right)\right]\)
\(=\left(y-z\right)\left[\left(x-z\right)\left(x-y\right)\right]\)
b \(x^8y^8+x^4y^4+1=x^8y^8+2x^4y^4+1-x^4y^4=\left(x^4y^4\right)^2+2x^4y^4+1-\left(x^2y^2\right)^2\)
\(=\left(x^4y^4+1\right)^2-\left(x^2y^2\right)^2=\left(x^4y^4-x^2y^2+1\right)\left(x^4y^4+x^2y^2+1\right)\)
c \(x^2y+xy^2+xz^2+x^2z+y^2z+yz^2+2xyz=\left(x^2y+x^2z+xyz+xy^2\right)+\left(xz^2+yz^2+xyz+y^2z\right)\)
\(=x\left(xy+xz+yz+y^2\right)+z\left(xz+yz+xy+y^2\right)=\left(x+z\right)\left(xy+xz+yz+y^2\right)\)
\(=\left(x+z\right)\left(x\left(y+z\right)+y\left(y+z\right)\right)=\left(x+z\right)\left(x+y\right)\left(y+z\right)\)
a \(3xyz+x\left(y^2+z^2\right)+y\left(x^2+z^2\right)+z\left(x^2+y^2\right)=3xyz+xy^2+xz^2+x^2y+yz^2+x^2z+y^2z\)
\(=\left(x^2y+x^2z+xyz\right)+\left(xy^2+xyz+y^2z\right)+\left(xyz+xz^2+yz^2\right)\)
\(=x\left(xy+xz+yz\right)+y\left(xy+xz+yz\right)+z\left(xy+xz+yz\right)=\left(x+y+z\right)\left(xy+xz+yz\right)\)
\(A=\left(x^2+y^2+z^2\right)\left(x+y+z\right)^2-\left(xy+yz+zx\right)^2\left(1\right)\)
Đặt \(x^2+y^2+z^2=a\)
\(xy+yz+zx=b\Rightarrow2\left(xy+yz+zx\right)=2b\)
\(\Rightarrow a+2b=\left(x+y+z\right)^2\)
Kết hợp (1) ta được : \(A=a\left(a+2b\right)+b^2\)
\(=a^2+2ab+b^2\)
\(=\left(a+b\right)^2\)
\(=\left(x^2+y^2+z^2+xy+yz+zx\right)^2\)
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)