Graph polynomial functions using zeros, factorizations, and end behavior.
Predict local zero behavior from multiplicity
Problem
For \(f(x)~=~(x~-~2)(x~+~1)^2\), describe the local behavior at every zero, including multiplicity, crossing or touching, and flattening.
Big Picture
What this problem is really about
Translate each factor exponent into three local questions: is the multiplicity odd or even, does the graph cross or turn, and is the multiplicity large enough to create noticeable flattening? Parity controls the sign change, while multiplicity size controls how flat the curve looks near the zero. Keeping those roles separate prevents one feature from being mistaken for another.
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