<s>[INST] How can $f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$ be factorized into a product of two polynomials? Let $x,y$ be 2 coprime integers. I assume the following polynomial:$$f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$$ is not irreducible. So there must be at least 2 other polynomials of degree $\leq 4$ such that: $$f(x,y)=g(x,y)h(x,y)$$
How can one find $g(x,y)$ and $h(x,y)$?
 [/INST] The important thing to notice
is that
$f(x, y)$
is homogeneous;
every term
of the form
$x^ay^b$
has the same value of
$a+b$.
Therefore,
dividing by
$y^n$,
where $n$ is this sum,
$\begin{array}\\
f(x,y)
&= x^4+x^3y+x^2y^2+xy^3+y^4\\
&= y^4 \left((x/y)^4+(x/y)^3+(x/y)^2+(x/y)+1 \right)\\
&= y^4 \left(r^4+r^3+r^2+r+1 \right)
\qquad\text{where } r=x/y\\
\end{array}
$
Now the problem is reduced
to factoring
$r^4+r^3+r^2+r+1
$.
</s>