read walker's algebraic curves, the first few chapters, for a nice discussion of this. there you will find I believe something like e.g. that a curve of degree d with more than (1/2)(d-1)(d-2) singular points is reducible, in the sense that its equation is a product of two equations, and a curve with more than (1/2)d(d-1) singular points is non reduced, in the sense that its equation is not square free. hence this is a stackexchange level question.
the reason for these facts is visible topologically if you know that a complex curve of degree d is topologically a specialization of a surface of genus g = (1/2)(d-1(d-2), and the maximum finite number of singularities occurs for a union of d lines.
these results are proved by the strong bezout theorem. e.g. if a cubic curve has two singularities, then the line through them meets the curve with multiplicity 2x2 = 4, hence lies inside the curve.
hence this is a stackexchange level question.
