Graph polynomial functions using zeros, factorizations, and end behavior.
List real zeros and integer multiplicities from factors
Problem
For \(f(x)~=~(x~-~4)^3(x~+~2)\), list the real zeros and multiplicities, then state the number of distinct real zeros, the sum of real multiplicities, the degree, and the number of nonreal zeros counted with multiplicity.
Big Picture
What this problem is really about
Separate the number of distinct zeros from the number counted with multiplicity. The bases of the linear factors locate the distinct real zeros, while their exponents contribute to the multiplicity total. Compare that total with the polynomial’s degree; any degree not already accounted for would have to come from nonreal zeros in conjugate pairs.
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