Extend polynomial identities to complex numbers for higher-degree polynomial work.
Factor a sum of squares over the complex numbers using conjugate imaginary factors
Problem
Factor \(x^{2}~+~9\) completely over the complex numbers.
Big Picture
What this problem is really about
A real sum of squares becomes a difference of squares over the complex numbers because the square of an imaginary multiple is negative. Rewrite the constant square as the negative square of that imaginary multiple, then apply the difference-of-squares identity. The resulting linear factors must be complex conjugates, whose imaginary terms cancel when multiplied.
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