What is the general rule for determining when a graph crosses the x-axis at an x-intercept or when the graph just touches and turns away from the x-axis?

Find this general rule by graphing the three functions below:

y = (x + 1)2(x - 2)

y = (x - 4)3(x - 1)2

y = (x - 3)2(x + 4)4

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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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