9. Find all binary polynomials $P$ that satisfy the following conditions:
(1) $P$ is a homogeneous polynomial of degree $n$;
(2) For all real numbers $a, b, c$, $P(a+b, c)+P(b+c, a)+P(c+a, b)=0$;
(3) $P(1,0)=1$.
9. We hope to use the so-called "substitution method" to find some factors of $p(x, y)$ from (2). For example, let $a=b=c=y$, we get $p(2 y, y)=$ 0. This indicates that when $x=2 y$, $p(x, y)=0$, hence $p(x, y)$ has a factor $x-2 y$. Again, let $a=b=y, c=-2 y$ we get $p(2 y,-2 y) + 2 p(-y, y)=0$, from (1) we have $(-2)^{n} p(-y, y) + 2 p(-y, y)=0$, so $n=1$ or $p(-y, y)=0$. If $n=1$, then $p(x, y)=k(x,-2 y)$, and from (3) we know $k=1$. If $p(-y, y)=0$, then $p(x, y)$ has a factor $x+y$. Let $p(x, y)=(x+y) p_{1}(x, y)$, substituting into (2), we can see that $p_{1}(x, y)$ still satisfies (2). If $n \neq 1$, then $p_{1}(x, y)=(x+y) p_{2}(x, y) \cdots$ and so on, eventually proving that $p(x, y)=(x+y)^{n-1}(x-2 y)$.