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A ) xy(z+y)+yz(y+z)+zx(z+x)
=y.[x(z+y)+z(y+z)]+zx(z+x)
=y.(xz+xy+zy+z2)+zx(z+x)
=y.(xz+z2+xy+zy)+zx(z+x)
=y.[z.(z+x)+y.(z+x)]+zx(z+x)
=y.(z+x)(z+y)+zx(z+x)
=(z+x)[y(z+y)+zx]
=(z+x)(yz+y2+zx)
B )xy(x+y)-yz(y+z)-zx(z-x)
=y.[x(x+y)-z(y+z)]-zx(z-x)
=y.(x2+xy-zy-z2)-zx(z-x)
=y.(x2-z2+xy-zy)-zx(z-x)
=y.[(x+z)(x-z)+y.(x-z)]-zx(z-x)
=y.(x-z)(x+z+y)+zx(x-z)
=(x-z)[y(x+z+y)+zx]
=(x-z)(yx+yz+y2+zx)
=(x-z)(yx+zx+yz+y2)
=(x-z)[x.(y+z)+y.(y+z)]
=(x-z)(y+z)(x+y)
b. \(\text{ xy(x+y)-yz(y+z)-xz(z-x) =xy(x+y+z-z)+yz(y+z)+xz(x-z) =xy(x-z)+xy(y+z)+yz(y+z)+xz(x-z) =(x+y)(y+z)(x-z) }\)
a: \(2x^2+3xy-14y^2\)
\(=2x^2+7xy-4xy-14y^2\)
\(=\left(2x^2+7xy\right)-\left(4xy+14y^2\right)\)
\(=x\left(2x+7y\right)-2y\left(2x+7y\right)\)
\(=\left(2x+7y\right)\left(x-2y\right)\)
b: \(\left(x-7\right)\left(x-5\right)\left(x-3\right)\left(x-1\right)+7\)
\(=\left(x-7\right)\left(x-1\right)\left(x-5\right)\left(x-3\right)+7\)
\(=\left(x^2-8x+7\right)\left(x^2-8x+15\right)+7\)
\(=\left(x^2-8x\right)^2+15\left(x^2-8x\right)+7\left(x^2-8x\right)+105+7\)
\(=\left(x^2-8x\right)^2+22\left(x^2-8x\right)+112\)
\(=\left(x^2-8x\right)^2+8\left(x^2-8x\right)+14\left(x^2-8x\right)+112\)
\(=\left(x^2-8x\right)\left(x^2-8x+8\right)+14\left(x^2-8x+8\right)\)
\(=\left(x^2-8x+8\right)\left(x^2-8x+14\right)\)
c: \(\left(x-3\right)^2+\left(x-3\right)\left(3x-1\right)-2\left(3x-1\right)^2\)
\(=\left(x-3\right)^2+2\left(x-3\right)\left(3x-1\right)-\left(x-3\right)\left(3x-1\right)-2\left(3x-1\right)^2\)
\(=\left(x-3\right)\left[\left(x-3\right)+2\left(3x-1\right)\right]-\left(3x-1\right)\left[\left(x-3\right)+2\left(3x-1\right)\right]\)
\(=\left(x-3+6x-2\right)\left(x-3-3x+1\right)\)
\(=\left(7x-5\right)\left(-2x-2\right)\)
\(=-2\left(x+1\right)\left(7x-5\right)\)
d: \(xy\left(x-y\right)+yz\left(y-z\right)+zx\left(z-x\right)\)
\(=x^2y-xy^2+y^2z-yz^2+zx\left(z-x\right)\)
\(=\left(x^2y-yz^2\right)-\left(xy^2-y^2z\right)+xz\left(z-x\right)\)
\(=y\left(x^2-z^2\right)-y^2\left(x-z\right)-xz\left(x-z\right)\)
\(=y\cdot\left(x-z\right)\left(x+z\right)-\left(x-z\right)\left(y^2+xz\right)\)
\(=\left(x-z\right)\left(xy+zy-y^2-xz\right)\)
\(=\left(x-z\right)\left[\left(xy-y^2\right)+\left(zy-zx\right)\right]\)
\(=\left(x-z\right)\left[y\cdot\left(x-y\right)-z\left(x-y\right)\right]\)
\(=\left(x-z\right)\left(x-y\right)\left(y-z\right)\)
a/ \(\left(x-y\right)\left(z-x\right)\left(z-y\right)\)
b/ \(\left(1-y\right)\left(y-x\right)\)
a. \(\left(x-y\right)\left(z-x\right)\left(z-y\right)\)
b. \(\left(1-y\right)\left(y-x\right)\)
a) xy(x + y) + yz(z + y) + zx(z + x) + 3xyz
= [xy(x + y) + xyz] + [yz(z + y) + xyz] + [zx(z + x) + xyz]
= xy(x + y + z) + yz(x + y + z) + zx(x + y + z)
= (xy + yz + zx)(x + y + z)
b) Vô câu hỏi tương tự
a) xy(x + y) + yz(z + y) + zx(z + x) + 3xyz
= [xy(x + y) + xyz] + [yz(z + y) + xyz] + [zx(z + x) + xyz]
= xy(x + y + z) + yz(x + y + z) + zx(x + y + z)
= (xy + yz + zx)(x + y + z)
b) tương tự
\(x\left(y^2-z^2\right)+y\left(z^2-x^2\right)+z\left(x^2-y^2\right)\)
\(=xy^2-xz^2+yz^2-x^2y+zx^2-zy^2\)
\(=xy^2-xz^2+yz^2-x^2y+zx^2-zy^2-xyz+xyz\)
\(=\left(yz^2-xz^2-xyz+x^2z\right)-\left(zy^2-xyz-xy^2+x^2y\right)\)
\(=z\left(yz-xz-xy+x^2\right)-y\left(zy-xz-xy+x^2\right)\)
\(=\left(z-y\right)\left(yz-xz-xy+x^2\right)\)
\(=\left(z-y\right)\left[y\left(z-x\right)-x\left(z-x\right)\right]\)
\(=\left(z-y\right)\left(y-x\right)\left(z-x\right)\)
xy(x+y)-yz(y+z)-zx(z-x)
=y.[x.(x+y)-z.(y+z)]-zx.(z-x)
=y.(x2+xy-zy-z2)-zx.(z-x)
=y.[(x-z)(x+z)-y.(z-x)]-zx.(z-x)
=y.[-(z-x)(x+z)-y.(z-x)]-zx.(z-x)
=y.(z-x)(-x-z-y)-zx.(z-x)
=(z-x)(-xy-zy-y2-zx)
=(z-x)[-x.(y+z)-y.(y+z)]
=(z-x)(y+z)(-x-y)
=-(z-x)(y+z)(x+y)
\(=xyz-xy-z\left(x+y\right)+x+y+z-1\)
\(=xy\left(z-1\right)-\left(x+y\right)\left(z-1\right)+z-1\)
\(=\left(z-1\right)\left(xy-x-y+1\right)\)
\(=\left(z-1\right)\left[x\left(y-1\right)-\left(y-1\right)\right]\)
\(=\left(x-1\right)\left(y-1\right)\left(z-1\right)\)
a) \(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)-8\)
\(=\left[\left(x+1\right)\left(x+4\right)\right]\left[\left(x+2\right)\left(x+3\right)\right]-8\)
\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)-8\)
\(=\left(x^2+5x+5\right)^2-1-8\)
\(=\left(x^2+5x+5\right)^2-3^2\)
\(=\left(x^2+5x+2\right)\left(x^2+5x+8\right)\)
b) \(xy\left(x-y\right)+yz\left(y-z\right)+zx\left(z-x\right)\)
\(=xy\left(x-y\right)+y^2z-yz^2+z^2x-zx^2\)
\(=xy\left(x-y\right)+z^2\left(x-y\right)-z\left(x-y\right)\left(x+y\right)\)
\(=\left(x-y\right)\left(xy+z^2-zx-yz\right)\)
\(=\left(x-y\right)\left[x\left(y-z\right)-z\left(y-z\right)\right]\)
\(=\left(x-y\right)\left(x-z\right)\left(y-z\right)\)
a) ( x + 1 )( x + 2 )( x + 3 )( x + 4 ) - 8
= [ ( x + 1 )( x + 4 ) ][ ( x + 2 )( x + 3 ) ] - 8
= ( x2 + 5x + 4 )( x2 + 5x + 6 ) - 8
Đặt t = x2 + 5x + 5
bthuc ⇔ ( t - 1 )( t + 1 ) - 8
= t2 - 1 - 8
= t2 - 9
= ( t - 3 )( t + 3 )
= ( x2 + 5x + 5 - 3 )( x2 + 5x + 5 + 3 )
= ( x2 + 5x + 2 )( x2 + 5x + 8 )
b) xy( x - y ) + yz( y - z ) + zx( z - x )
= x2y - xy2 + y2z - yz2 + zx( z - x )
= ( y2z - xy2 ) - ( yz2 - x2y ) + zx( z - x )
= y2( z - x ) - y( z2 - x2 ) + zx( z - x )
= ( z - x )( y2 + zx ) - y( z - x )( z + x )
= ( z - x )( y2 + zx - yz - yx )
= ( z - x )[ ( y2 - yx ) - ( yz - zx ) ]
= ( z - x )[ y( y - x ) - z( y - x ) ]
= ( z - x )( y - x )( y - z )