Đặt \(\left\{{}\begin{matrix}a=x+y\\b=y+z\\c=x+z\end{matrix}\right.\Leftrightarrow x+y+z=\dfrac{a+b+c}{2}\)

\(8\left(x+y+z\right)^3-\left(x+y\right)^3-\left(y+z\right)^3-\left(z+x\right)^3\\ =8\left(\dfrac{a+b+c}{2}\right)^3-a^3-b^3-c^3\\ =\left(a+b+c\right)^3-a^3-b^3-c^3\\ =\left(a+b\right)^3+c^3+3\left(a+b\right)c\left(a+b+c\right)-\left(a+b\right)^3+3ab\left(a+b\right)-c^3\\ =3\left(a+b\right)\left(ac+bc+c^2+ab\right)\\ =3\left(a+b\right)\left(b+c\right)\left(a+c\right)\\ =3\left(x+y+y+z\right)\left(y+z+z+x\right)\left(z+x+x+y\right)\\ =3\left(x+2y+z\right)\left(x+y+2z\right)\left(2x+y+z\right)\)

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 )

a) \([(x-y)3 + (y-z)3]+ (z-x)3\)=\(\left(x-y+y-z\right)\left[\left(x-y\right)^2-\left(x-y\right)\left(y-z\right)+\left(y-z\right)^2\right]-\left(x-z\right)^3\)

\(=\left(x-z\right)\left[\left(\left(x-y\right)^2-\left(x-y\right)\left(y-z\right)+\left(y-z\right)^2-\left(x-z\right)^2\right)\right]\)

\(=\left(x-z\right)\left[\left(x-y\right)\left(x-y-y+z\right)+\left(y-z-x+z\right)\left(y-z+x-z\right)\right]=\left(x-z\right)\left[\left(x-2y+z\right)\left(x+z\right)-\left(x-y\right)\left(x+y-2z\right)\right]\)

\(=\left(x-z\right)\left(x-y\right)\left(x-2y+z-x-y+2z\right)=\left(x-z\right)\left(x-y\right)\left(z-y\right)3\)

b) \(=y^2\left(x^2y-x^3+z^3-z^2y\right)-z^2x^2\left(z-x\right)=y^2\left[-y\left(z^2-x^2\right)-\left(z^3-x^3\right)\right]-z^2x^2\left(z-x\right)\)

\(=y^2\left(z-x\right)\left(-yz-xy-z^2-zx-x^2\right)-z^2x^2\left(z-x\right)=\left(z-x\right)\left(-y^3z-xy^2-z^2y^2-xyz-x^2y^2-z^2x^2\right)\)

đến đây coi như là thành nhân tử rồi nha. em muốn gọn thì ráng ngồi nghĩ rồi tách nha. chỉ cần nhóm mấy cái có ngoặc giống nhau là đc. k khó đâu. chịu khó nghĩ để rèn luyện nha

c) \(x^8+2x^4+1-x^4=\left(x^4+1\right)^2-x^4=\left(x^4+1-x^2\right)\left(x^4+1+x^2\right)\)

\(\left(9a^3-6a^2\right)+\left(6a^2-4a\right)+\left(-9a+6\right)=3a^2\left(3a-2\right)+2a\left(3a-2\right)-3\left(3a-2\right)=\left(3a-2\right)\left(3a^2+2a-3\right)\)

d) em sửa đề đi. đề sai rồi. đồng nhất hệ số phải có dấu bằng nha.

có gì liên hệ chị. đúng nha ;)

a: =(x-z)(y+8)

b; =x^2-2x-3x+6

=(x-2)(x-3)

c: =x^4+10x^2-x^2-10

=(x^2+10)(x^2-1)

=(x^2+10)(x-1)(x+1)

-(z+x)³  mới đúng-

đặt x+y=a , y+z=b , z+x=c thì a+b+c=2(x+y+z)

ta có 8(x+y+z)³-(x+y)³-(y+z)³-(z+x)³=[2(x+y+z)]³-(x+y)³-(y+z)³-(z+x)³=(a+b+c)³-a³-b³-c³=3(a+b)(b+c)(c+a) 

=3(x+2y+z)(y+2z+x)(z+2x+y)

1) \(x^2-4xy+4y^2+xz-2yz\)

\(=\left(x^2-4xy+4y^2\right)+\left(xz-2yz\right)\)

\(=\left(x-2y\right)^2+z\left(x-2y\right)\)

\(=\left(x-2y\right)\left(x-2y+z\right)\)

2) \(\left(x-y\right)^3+\left(x+y\right)^3\)

\(=\left[\left(x-y\right)+\left(x+y\right)\right]\left[\left(x-y\right)^2-\left(x-y\right)\left(x+y\right)+\left(x+y\right)^2\right]\)

\(=\left(x-y+x+y\right)\left(x^2-2xy+y^2-x^2+y^2+x^2+2xy+y^2\right)\)

\(=2x\left(x^2+3y^2\right)\)