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

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

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

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

=> đpcm

=.= hok tốt!!

Answer 2

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

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

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

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

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

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

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

=> ĐPCM

Answer 3

ta có: \(\left(x+y+z\right)^3-x^3-y^3-z^3=\left[\left(x+y\right)+z\right]^3-x^3-y^3-z^3.\)

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

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

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

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

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