For polynomials, the graph will cross the x-axis if the multiplicity of the real root is odd, and just touch the x-axis if the multiplicity of the real root is even. (The multiplicity of the root is the number of times it occurs as a root)

(a) `y=(x+1)^2(x-2)` The graph crosses at x=2 (multiplicity 1) but touches at x=-1 (mulitplicity 2)

(b) `y=(x-4)^3(x-1)^2` The graph crosses at x=4 (multiplicity 3) but touches at x=1 (m=2)

(c) `y=(x-3)^2(x+4)^4` The graph touches at x=3 and x=-4 as the multiplicities are both even.

The graphs: (a) black, (b) red, (c) green

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