Like the graphic in Path One, the left side of above graphic shows the swallowtail in parametric space, and the right side shows the polynomial represented by the red point.
The asymmetric shape of the polynomials with the root at x != 0 makes it clear that this root cannot be a quadruple root. Therefore, the root must be a simple double root.
In the case of the polynomial with root at x = 0, we cannot determine the multiplicity of the root from the shape of the graph. If the root at x = 0 were a quadruple root, then the polynomial would be p(x) = x4. However, since (a, b, c) != (0, 0, 0), the polynomial is not p(x) = x4, and therefore its root at zero is not a quadruple root. This polynomial has another critical point, whereas p(x) = x4 has only one critical point, the quadruple root. Therefore, the root at x = 0 is a double root.
To see a path along the intersection curve (which includes (a, b, c) = (0, 0, 0)), see Path Three.