Toán Lớp 8: CMR: (x+y+z)^3 = x^3 + y^3 + z^3 + 3(x+y)(y+z)(z+x)

Question

Toán Lớp 8:  CMR: (x+y+z)^3 = x^3 + y^3 + z^3 + 3(x+y)(y+z)(z+x)

huenthanh97 · Answer

~rai~        ((x+y+z)^3=x^3+y^3+z^3+3(x+y)(y+z)(z+x)   text{X&eacute;t VT=}(x+y+z)^3   quad quad quad=[(x+y)+z]^3   quad quad quad=(x+y)^3+3(x+y)^2.z+3(x+y).z^2+z^3   quad quad quad=x^3+3x^2y+3xy^2+y^3+3z(x^2+2xy+y^2)+3xz+3yz+z^3   quad quad quad=x^3+y^3+z^3+3x^2y+3xy^2+3x^2z+6xyz+3y^2z+3xz^2+3yz^2     quad quad quad=x^3+y^3+z^3+(3xy^2+3xy^2)+(3x^2z+3xyz)+(3xyz+3y^2z)+(3xz^2+3yz^2)   quad quad quad=x^3+y^3+z^3+3xy(x+y)+3xz(x+y)+3yz(x+y)+3z^2(x+y)   quad quad quad=x^3+y^3+z^3+3(x+y)(xy+xz+yz+z^2)   quad quad quad=x^3+y^3+z^3+3(x+y)[(xy+xz)+(yz+z^2)]   quad quad quad=x^3+y^3+z^3+3(x+y)[x(y+z)+z(y+z)]   quad quad quad=x^3+y^3+z^3+3(x+y)(y+z)(z+x)   quad quad quad=VP.   Rightarrow (x+y+z)^3=x^3+y^3+z^3+3(x+y)(y+z)(z+x). quad(đpcm) )

dongoclandiem · Answer

(x+y+z)^3 = x^3 + y^3 + z^3 + 3(x+y)(y+z)(z+x)     (x+y+z)^3 = x^3 + y^3 +z^3 +3xyz +3xz^2 +3yx^2 +3zx^2 +3xy^2 +3xyz+3zy^2+3yz^2     (x+y+z)^3 = x^3 + 3x^2 y+3y^2 x+y^3 +3x^2 z+6xyz+3y^2 z+3xz^2 +z^3     (x+y+z)^3 = (x+y)^3 +3z(x+y)^2 +3z^2 (x+y) +z^3     (x+y+z)^3 = (x+y+z)^3     => (x+y+z)^3 = x^3 + y^3 + z^3 + 3(x+y)(y+z)(z+x)