
Exercieses on form
Hello.
I am reading Fulton's book "Algebraic Curves" and I'd like to see how are solved exercises 2.34 and 2.35. They say:
1) Factor $\displaystyle Y^22XY^2 + X^3$ into linear factors in $\displaystyle C[X,Y]$ (where C denotes complex numbers).
Just above it is stated that: "Up to powers of $\displaystyle X_{n+1}$, factoring a form $\displaystyle F \in R[X_1, . . . ,X_{n+1}]$ is the same as
factoring $\displaystyle F \in R[X_1, . . . ,X_n].$ In particular, if $\displaystyle F \in k[X,Y ] $is a form, k algebraically closed, then F factors into a product of linear factors."
It must be helpful... but I don't know how to use it. And also, "linear factors" in $\displaystyle C[X,Y]$ are of the form $\displaystyle XaY$?
Thank you for your help.