Graph polynomial functions using zeros, factorizations, and end behavior.
Inventory polynomial graph features from factored form
Problem
For \(f(x)~=~(x~-~2)(x~+~1)^2\), list each real zero with its multiplicity and crossing or touching behavior, state the degree and leading coefficient, compute the \(y\text{-}\text{intercept}\), and describe \(\text{both}~\text{ends}\).
Big Picture
What this problem is really about
Read a factored polynomial as a map of its graph. Each factor gives a zero, and the exponent’s parity tells whether the curve crosses or merely touches the axis there; the total degree and leading sign control the tails. Finally, substitute zero for the vertical intercept and check that all of these features fit one continuous curve.
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