I am looking for factorization of polynomials of several variables in the way outlined below.


Consider a second degree polynomial of two variables over the complex numbers.


"P(x,y) = Ax^2 + Bxy + Cy^2 + Dx + Ey + F" (see the edit below)


Experimenting with some polynomials of this sort showed me that factorization is possible in the following way.



```
     "P(x,y) = (ax + by + c)(dx + ey + f)"  (see the edit below) ,

```

the coefficients being over the complex numbers. 


So, given an nth degree polynomial in n variables without a constant term, is it always possible to factorize it into n linear factors each having n variables in the above way? (This rings bells about the fundamental theorem of algebra). 


Please suggest a reading material or journal, if any.


[EDIT: I am sorry, I erred. I have edited my question. The edit is that the polynomial has no constant term:  

 P(x,y) = Ax^2 + Bxy + Cy^2 + Dx + Ey  

and in the expected factorization, the last linear factor does not have a constant term, too:  

 P(x,y) = (ax + by + c)(dx + ey + f) ]


I kept the original question as it is for documentation purposes. 



Factorizing polynomials of several variables (in a different perespective)
Let K be a field. The ring $K[X\_1,...,X\_n]$ is factorial, which means that any polynomial in n variables can be factored into a product of irreducible polynomials. But of course, these polynomials are not of degree 1 in general. If a polynomial can be factored as a product of terms of degree one, then its zero set is a finite union of hyperplanes. An interesting family of irreducible polynomials in C[X,Y] is given by $Y^2-X(X-1)(X-L)$, $L$ different from 0 and 1. The zero set in $P^2(C)$ is called an elliptic curve and it is diffeomorphic to a torus $S^1\times S^1$.


As a reference, I can point to Lang "Algebra", or Jacobson "Basic algebra".


