Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Decide whether a proposed algebraic identity is true for complex numbers
Problem
Is this identity true for complex numbers? \((z+w)^2=z^2+2\text{zw}+w^2\)
Big Picture
What this problem is really about
Polynomial identities extend to complex numbers because the same distributive and commutative laws apply. We’ll rewrite the square as a product, distribute all four terms, recognize both cross products, combine them, and compare the expansion term by term with the proposed identity.
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