Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Factor polynomials completely over the complex numbers using the sum-of-squares identity
Problem
Completely factor \((x^{2}+4)(x-1)\) over \(ℂ\). Split \(x^{2}+4\) into conjugate linear factors and retain the existing factor \(x-1\).
Big Picture
What this problem is really about
Complete factorization preserves every existing linear factor while splitting any remaining quadratic over the stated field. We’ll retain the original factor, rewrite the sum-of-squares quadratic as conjugates, restore the full product, and check that the number of linear factors matches the polynomial’s degree.
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