(1)(a-b\()^{3}\)+(b-c\()^{3}\)+(c-a\()^{3}\)
=(a-b+b-c)[(a-b\()^{2}\)-(a-b)(b-c)+(b-c\()^{2}\)]
-(a-c\()^{3}\)
=(a-c)[(a-b\()^{2}\)-(a-b)(b-c)+(b-c\()^{2}\)-(a-c\()^{2}\)]
=3(a-b)(b-c)(c-a)

(2)
(x+y+z\()^{3}\)-\(x^{3}\)-\(y^{3}\)-\(z^{3}\)
=(x+y+z-x)[(x+y+z\()^{2}\)+x(x+y+z)+\(x^{2}\)]-(y+z)(\(y^{2}\)-yz+\(z^{2}\))
=(y+z)(\(x^{2}\)+\(y^{2}\)+\(z^{2}\)+2xy+2yz+2xz+\(x^{2}\)+xy+xz+\(x^{2}\))
-(y+z)(\(y^{2}\)-yz+\(z^{2}\))
=(y+z)(3\(x^{2}\)+3xy+3yz+3xz)
=(y+z)[3x(x+y)+3z(x+y)]
=3(x+y)(y+z)(z+x)

(3)\(a^{3}\)+\(b^{3}\)+\(c^{3}\)-3abc
=(a+b\()^{3}\)-3ab(a+b)+\(c^{3}\)-3abc
=(a+b+c)[(a+b\()^{2}\)-c(a+b)+\(c^{2}\)]-3ab(a+b+c)
=(a+b+c)(\(a^{2}\)+\(b^{2}\)+\(c^{2}\)-ab-bc-ca)