let $f_r$ an homogenous form of degree $r$ and $f_{r+1}$ a form of degree $r-1$ without commun factors, in $k[x_1, \dots, x_n]$. I want to show that the sum $f = f_r + f_{r+1}$ is irreducible. (exercise 2.34 from Fulton, Algebraic Curves.)
I tried to homogenize the polynomial and I get $f^* = x_{n+1}f_r + f_{r+1}$. I know that $f$ factors $\Leftrightarrow$ $f^*$ does (except for factorisation of $f^*$ by the power of $x_{n+1}$).
But I have no more information with this. Can I get some hints ? Thanks.
If $f=gh$, with $ord(g),ord(h)>0$ then $ord(g)+ord(h)=r-1$ and $deg(g)+deg(h)=r$.
We have that $$r-1=ord(g)+ord(h)\leq deg(g)+ord(h)\leq deg(g)+deg(h)=r.$$ Therefore one of the two inequalities is an equality. This means that either $g$ or $h$ is homogeneous, and therefore both are homogeneous, together with $f$.