How does root multiplicity effect the behavior of a function?

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I understand that the multiplicity of a real zero determines whether the function touches or passes through the x-axis, but I’m lost as to WHY the multiplicity affects this characteristic. I can easily test a point in each interval of the function to see that the idea works perfectly, but I’m just curious if there is a concise explanation as to why an even multiplicity makes the function bounce, while an odd multiplicity makes the function pass through the x-axis. Am I overthinking things, or can someone help me out?

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SporkOfDesire

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Posted: 4 months ago
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