1. (1) Use the Factor Theorem to prove that $a-b, b-c, c-a$ are all factors of $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)$.
(2) Using the conclusion from (1), factorize $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)$.
(3) Factorize: $(x+y+z)^{3}-x^{3}-y^{3}-z^{3}$.
1. (1) omitted
(2) From (1), we know $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)=k(a-b)(b-c)(c-a)$. Let $a=2, b=1, c=0$, we get $k=-1$. Therefore, $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-\dot{b})=-(a-b)(b-c)(c-a)$.
(3) $3(x+y)(y+z)(z+x)$.