Question

\(\text{Phân tích đa thức thành nhân tử:}\)

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

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

 

 

Answer 1

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

\(=x^3+y^3+z^3+3x^2y+3x^2z+3y^2z+3xy^2+3xz^2+3yz^2+6xyz-x^3-y^3-z^2\) 

\(=3x^2y+3xy^2+3x^2z+3xz^2+3y^2z+3yz^2+6xyz\)

\(=3xy\left(x+y\right)+3xz\left(x+z\right)+3yz\left(y+z\right)+6xyz\)

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

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

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

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

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

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

Answer 2

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

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

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

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

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

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

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

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

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

Answer 3

Em bê đê đó anh, ờ hớ hớ