Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Simplify a product of complex conjugates
Problem
Evaluate \((3+4i)(3-4i)\) as a conjugate product. Use \(a^{2}+b^{2}\) or expand and simplify \(i^{2}\) to obtain the real result.
Big Picture
What this problem is really about
A complex number times its conjugate is real because the imaginary cross terms cancel. We’ll identify the shared real and opposite imaginary parts, apply the sum-of-squares identity, evaluate the two component squares, and confirm that the imaginary coefficient is zero.
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