Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Factor a sum of squares over the complex numbers using \(a^2+b^2=(a+bi)(a-bi)\)
Problem
Factor \(a^{2}+16\) over the complex numbers by expressing the constant term through \((4i)^{2}\) and applying difference of squares.
Big Picture
What this problem is really about
A real sum of squares becomes factorable over the complex numbers because the imaginary unit turns one square negative. We’ll express the constant as the square of an imaginary term with the required sign, recognize a difference of squares, and form conjugate linear factors. Multiplying those factors back cancels the imaginary terms and verifies the original sum.
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