Question

CMR: nếu x + y + z + t = 0  thì x^3 + y^3 + z^3 + t^3 = 3.(xy-zt).(z+t)

Answer 1

\(x+y+z+t=0\Rightarrow t=-\left(x+y+z\right)\)

Ta có: 

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

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

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

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

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

Do VT = VP nên ta có đpcm.