Identify zeros from factorizations and use them to sketch polynomial graphs.
Write the unique least-degree factored polynomial with a specified leading coefficient
Problem
Write the least-degree factored polynomial with leading coefficient \(1\) that crosses at \(x~=~-2\) and touches at \(x~=~3\).
Big Picture
What this problem is really about
Build the polynomial from the graph one intercept at a time. A crossing needs an odd multiplicity and a touch needs an even multiplicity; because the problem asks for least degree, use the smallest positive exponent with the required parity. Then multiply the factors and check the leading coefficient, adding a scale factor only if needed.
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