Graph polynomial functions using zeros, factorizations, and end behavior.
Use a known zero to completely factor and verify a polynomial
Problem
Given that \(1\) is a zero of \(f(x)~=~x^3~-~6x^2~+~11x~-~6\), factor \(f\) completely and list all real and nonreal zeros with multiplicity.
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What this problem is really about
Turn the known zero into its corresponding linear factor, then divide that factor out of the polynomial. A zero remainder confirms the first step, but the job is not finished until the quotient is factored completely and every zero is classified. Multiply the factors back together and compare degrees to verify both the reconstruction and the complete zero inventory.
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