Singular points of an irreducible affine plane cubic
= Singular points of an irreducible affine plane cubic
An irreducible affine plane cubic has at most one singular point. If two existed, restriction of its cubic polynomial to their joining line would have a zero of multiplicity at least two at each point. A polynomial of degree at most three cannot have those four zeros unless it vanishes identically, making the line a component and contradicting irreducibility.