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Unformatted text preview: 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 2. 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 ?
11. Here are a few other questions for you to discuss with your classmates.
(a) How many local extrema could a polynomial of degree n have? How few local extrema...
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