Question

Chứng minh rằng: \(\left(x+y+z\right)^3-x^3-y^3-z^3=3\left(x+y\right)\left(y+z\right)\left(z+x\right)\)

Answer 1

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

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

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

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

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

\(=3\left(x+y\right)\left(y+z\right)\left(z+x\right)=VP\left(\text{ĐPCM}\right)\)