Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Verify a sum-of-squares factorization over the complex numbers
Problem
Expand \((x+3i)(x-3i)\). Show the middle terms cancel and use \(i^{2}=-1\) to verify the resulting polynomial.
Big Picture
What this problem is really about
Verification requires expanding the conjugate factors all the way to an ordinary polynomial. We’ll write all four products, cancel the opposite imaginary middle terms, replace i squared with negative one, and compare the simplified result with the original sum of squares.
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