Extend polynomial identities to complex numbers for higher-degree polynomial work.
Find all complex roots of an already factored polynomial, including roots from a quadratic that gives a conjugate imaginary pair
Problem
Find every complex root of \((x~-~3)(x^{2}~+~4)~=~0\).
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What this problem is really about
Apply the zero-product property to the given factorization and solve every factor independently. A quadratic equation of the form x squared equals a negative real number contributes both imaginary square roots, not just one. Combine those roots with the roots from the linear factors and check that multiplicities account for the polynomial's degree.
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