There is one subtlety about the role of multiplicity that we need to discuss further; specifically we need to see 'how' the graph crosses the x-axis at a zero of odd multiplicity. In the section, we deliberately excluded the function f(x)=x from the discussion of the end behavior of f(x)=xn for odd numbers n and we said at the time that it was due to the fact that f(x)=x didn't fit the pattern we were trying to establish. You just showed in the previous exercise that the end behavior of a linear function behaves like every other polynomial of odd degree, so what doesn't f(x)=x do that g(x)=x3 does? It's the 'flattening' for values of x near zero. It is this local behavior that will distinguish between a zero of multiplicity 1 and one of higher odd multiplicity. Look again closely at the graphs of a(x)=x(x+2)2 and F(x)=x3(x+2)2 from Exercise 3.1.1. Discuss with your classmates how the graphs are fundamentally different at the origin. It might help to use a graphing calculator to zoom in on the origin to see the different crossing behavior. Also compare the behavior of a(x)=x(x+2)2 to that of g(x)=x(x+2)3 near the point (−2,0). What do you predict will happen at the zeros of f(x)=(x−1)(x−2)2(x−3)3(x−4)4(x−5)5?