Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Factor a sum of squares of the form \(x^2+k\) over the complex numbers
Problem
Factor \(x^{2}+4\) over \(ℂ\) by rewriting it as \(x^{2}-(2i)^{2}\) and applying the difference-of-squares pattern.
Big Picture
What this problem is really about
Over the complex numbers, a sum of squares can be rewritten as a difference involving an imaginary square. We’ll identify the square root of the constant, attach i, apply the difference-of-squares pattern, and multiply the conjugates back to confirm the sign.
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