Is $4x^2+3xy^2+y^3+7$ irreducible in $\mathbb{C}[x,y]$?

Question

Is $4x^2+3xy^2+y^3+7$ irreducible in $\mathbb{C}[x,y]$?

I tried to group it like $(4)x^2+(3y^2)x+(y^3+7)$. This is a polynomial with degree $2$ so I am thinking of applying quadratic formula... where do I go from there?

Answer 1

What would happen if your polynomial factored as a product $fg$? Let's look at the $x$-degrees of $f$ and $g$. If one of their degrees as a polynomial in $x$ is zero, then that polynomial must divide $4$, and is thus constant. So there are no non-trivial factorizations like this.

The only remaining possibility is that $f$ and $g$ are both linear in $x$. Then you may use the quadratic formula to figure out what the constant terms of $f$ and $g$ must be. If they're not in $\mathbb{C}[y]$, then the polynomial is irreducible.