Identify zeros from factorizations and use them to sketch polynomial graphs.
Use a known zero to factor a numeric polynomial and identify graph features
Problem
Given that \(2\) is a zero of \(p(x)~=~x^3~-~4x^2~+~x~+~6\), factor \(p(x)\) completely and state the zero behavior and end behavior.
Big Picture
What this problem is really about
Use the known zero to unlock the rest of the polynomial. The Factor Theorem turns that zero into a linear divisor; synthetic division lowers the degree, making the remaining quotient easier to factor. Once the complete factorization is available, its zeros, multiplicities, degree, and leading sign organize every requested graph feature.
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