Question

Phân tích đa thức sau thành nhân tử : (x+y+z)^3 -x^3-y^3-z^3

(Ko làm tắt giùm ạ)

Answer 1

( x + y + z )³ - x³ - y³ - z³

= [ ( x + y + z )³ - x³ ] - ( y³ + z³ )

= ( x + y + z - x )[ ( x + y + z )² + ( x + y + z )x + x² ] - ( y + z )( y² - yz + z² )

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

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

= ( y + z )( 3x² + 3yz + 3zx + 3xy )

= 3( y + z )( x² + yz + zx + xy )

= 3( y + z )[ ( x² + zx ) + ( xy + yz ) ]

= 3( y + z )[ x( x + z ) + y( x + z ) ]

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

Answer 2

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

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

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

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

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

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

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

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

\(=\left(y+z\right).\left[3x.\left(x+y\right)+3z.\left(y+x\right)\right]\)

\(=\left(y+z\right).\left(x+y\right).\left(3x+3z\right)\)

\(=3.\left(y+z\right).\left(x+y\right).\left(x+z\right)\)