Know the Fundamental Theorem of Algebra and connect it to polynomial roots.
Relate concrete complete factorizations over R and C
Problem
Give the complete factorization and roots of \(x^{2}~+~9\) over the real numbers and over the complex numbers.
Big Picture
What this problem is really about
Factorization depends on the number system allowed for the coefficients. A sum-of-squares quadratic has no real zeros and therefore remains irreducible over the reals, but solving the same zero equation over the complex numbers produces an imaginary conjugate pair. Convert those roots to linear factors and multiply them to verify the original positive constant.
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