Answer
please see "step by step" for sample explanation.
Work Step by Step
See Section 3-2 (page 356) on Turning Points of Polynomial Functions.
(If f is degree n, the graph has at most n-1 turning points)
Also see The Leading Coefficient Test (p. 350, end behavior )
( Odd $(\swarrow,\nearrow)$ or $(\nwarrow,\searrow)\quad$Even: $(\nwarrow,\nearrow)$ or $(\swarrow, \searrow)$ )
Also, see Multiplicity and x-lntercepts
(odd: crosses the x-axis, even: touches and turn)
... graphs tend to flatten out near zeros with multiplicity greater than one.
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Sample explanation:
n=20 is an even degree. Two cases of end behavior are possible.
1. Say the end behavior is $(\nwarrow,\nearrow)$.
By crossing the x axis at some point, it descends, $\searrow$, and, to satisfy the far right end behavior $(\nearrow)$ it must turn at some point and cross the x-axis again.
2. If the end behavior is $(\swarrow, \searrow)$
By crossing the x axis at some point, it rises, $\nearrow$, and, to satisfy the far right end behavior $(\searrow)$ it must turn at some point and cross the x-axis again.
So, in either case, if it crosses once, it has cross x at another point as well.
(Note: it can "touch and turn", but that wasn't the question. The question specifies CROSSING the x-xis, having odd multiplicity)
