when I say mutiplicity, I mean how many times x=a appears as a factor. E.G. in the polynomial x^2 + 2x + 1 = 0 , when factored we have (x + 1)(x + 1) = 0 and the roots are x = -1 and x = -1 again so this has a root of even multiplicity and therefore the graph touches the x-axis but does not cross it at x = -1.
I know what you mean. It sounds to me like you just described what is going on in your own words....