Extend polynomial identities to complex numbers for higher-degree polynomial work.
Compare complete factorization over R and C
Problem
Give the complete factorization and roots of \(x^{2}~+~4\) over the real numbers and over the complex numbers.
Big Picture
What this problem is really about
The coefficient field determines what counts as a complete factorization. Over the reals, a quadratic with negative discriminant remains irreducible; over the complex numbers, its imaginary roots produce conjugate linear factors. Multiplying the conjugate factors back together verifies that the imaginary middle terms cancel and the original real quadratic returns.
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