Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Solve \(u^2+b^2=0\) by factoring into complex conjugates
Problem
Solve \(x^{2}+9=0\) by factoring \(x^{2}+9\) into complex conjugate linear factors and applying the zero-product property to \(\text{both}~\text{factors}\).
Big Picture
What this problem is really about
Factoring over the complex numbers turns the sum of squares equation into two conjugate linear equations. We’ll form the factors, apply the zero-product property to both, solve each branch, and square the resulting pair to verify the complete solution set.
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