Sửa đề: \(x^3+y^3+z^3-3xyz\)

\(=\left(x+y\right)^3+z^3-3xy\left(x+y\right)-3xyz\)

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

\(=\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2-3xy\right)\)

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

  Ta có: 
x³ + y³ + z³ - 3xyz = (x+y)³ - 3xy(x-y) + z³ - 3xyz 
= [(x+y)³ + z³] - 3xy(x+y+z) 
= (x+y+z)³ - 3z(x+y)(x+y+z) - 3xy(x-y-z) 
= (x+y+z)[(x+y+z)² - 3z(x+y) - 3xy] 
= (x+y+z)(x² + y² + z² + 2xy + 2xz + 2yz - 3xz - 3yz - 3xy) 
= (x+y+z)(x² + y² + z² - xy - xz - yz). 

Ta có: 
x³ + y³ + z³ - 3xyz = (x+y)³ - 3xy(x-y) + z³ - 3xyz 
= [(x+y)³ + z³] - 3xy(x+y+z) 
= (x+y+z)³ - 3z(x+y)(x+y+z) - 3xy(x-y-z) 
= (x+y+z)[(x+y+z)² - 3z(x+y) - 3xy] 
= (x+y+z)(x² + y² + z² + 2xy + 2xz + 2yz - 3xz - 3yz - 3xy) 
= (x+y+z)(x² + y² + z² - xy - xz - yz). 
~~~~~~~~ 
Bài làm trên mình đã sử dụng hằng đẳng thức đáng nhớ sau: 
(a+b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a-b) 
=> a³ + b³ = (a+b)³ - 3ab(a-b). 
Chúc bạn học giỏi!

x³ + y³ + z³ - 3xyz = (x + y)³ - 3xy(x + y) + z³ - 3xyz 
= (x + y)³ + z³ - 3xy(x + y + z) 
= (x + y + z)³ - 3(x + y + z)(x + y)z - 3xy(x + y + z) 
= (x + y + z)³ - 3(x + y + z)(xy + yz + zx) 
= (x + y + z)[(x + y + z)² - 3xy - 3yz - 3zx)] 
= (x + y + z)(x² + y² + z² - xy - yz - zx)

k nha!

\(x^3+y^3+z^3+3xyz\)

\(=\left(x+y\right)^3-3xy\left(x+y\right)+z^3+3xyz\)

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

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

\(=\left(x+y+z\right)^3-3\left(x+y+z\right)\left(xy+yz+xz\right)\)

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

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

Trần Đức Thắng sai rùi X^3+y^3+z^3+3xyz cơ mà có phải X^3+y^3+z^3-3xyz đâu mà làm vậy 

\(x^3+y^3+z^3-3xyz\)

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

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

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

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

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

= (x + y)³ + z³ – 3x2y – 3xy2 - 3xyz 

= (x + y +z)[(x + y)² – (x + y)z + z²)] - 3xy(x + y + z)

= (x + y + z)(x² +2xy + y² – xz – yz +z² – 3xy)

= (x + y + z)(x² + y² +z² – xy - yz – xz)

x³ - y³ - z³ +3xyz

= (x³ - 3x²y +3xy² -y^3) +3x²y-3xy² - z³ +3xyz 

= [(x-y)³ -z³] + 3x²y -3xy² +3xyz

= (x-y-z)(x² + 2xy+y² +zx+zy + z²) + 3xy( x-y+z)

x3−y3−z3+3xyz=(x+y+z)(xy+yz+xz−x2−y2−z2) =-(x^3+y^3+y^3-3xyz)$

Ta tính x3+y2+z3−3xyz trước

ta có:

x3+y3+z3−3xyz=(x+y)3+z3−3xy(x+y)−3xyz=(x+y+z)(x2+y2+z2−xy−yz−xz)

=>x3−y3−z3+3xyz=(x+y+z)(xy+zy+xz−x2−y2−z2)

tick đúng đi chứ ! Làm mất công rồi !

\(\left(3x+1\right)^2-\left(3x-1\right)^2\)

\(=\left(3x+1-3x+1\right)\left(3x+1+3x-1\right)\)

\(=2\cdot6x\)

\(=12x\)

_________

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

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

\(=2x\cdot2y\)

\(=4xy\)

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

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

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

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

______

\(x^3+y^3+z^3-3xyz\)

\(=\left(x+y\right)^3-3xy\left(x-y\right)+z^3+3xyz\)
\(=\left[\left(x+y\right)^3+z^3\right]-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)^3-3z\left(x+y\right)\left(x+y+z\right)-3xy\left(x-y-z\right)\)
\(=\left(x+y+z\right)\left[\left(x+y+z\right)^2-3z\left(x+y\right)-3xy\right]\)
\(=\left(x+y+z\right)\left(x^2+y^2+z^2+2xy+2xz+2yz-3xz-3yz-3xy\right)\)
\(=\left(x+y+z\right)\left(x^2+y^2-xy-xz-yz\right)\)

\(=\left(x^3+y^3\right)+z^3-3xyz=\left(x+y\right)^3+z^3-3xy\left(x+y\right)-3xyz\)

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

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

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