Show that the polynomial $Y^2+X^2(X-1)^2 \in \Bbb R[X,Y]$ is irreducible.

Question

I'm studying basic algebraic geometry (from Fulton's Algebraic Curves). I'm required to show that the polynomial $$Y^2+X^2(X-1)^2 \in \Bbb R[X,Y]$$ is irreducible. I'm not that familiar with multivariable polynomials. How does one in general tackle such problems?

Answer 1

In general there are many ways. In this particular case the degree in $Y$ is low, so it is not hard to try to factor the polynomial 'by brute force'. If the polynomial factors, then it factors into two factors that are linear in $Y$, meaning that there exist $a(X),b(X),c(X),d(X)\in\Bbb{R}[X]$ such that $$Y^2+X^2(X-1)^2=(a(X)Y+b(X))(c(X)Y+d(X)),$$ which quickly leads to a contradiction.