Extend polynomial identities to complex numbers for higher-degree polynomial work.
Verify a complex zero of a real-coefficient polynomial using conjugate factors
Problem
Verify \(x~=~i\) as a zero of \(x^{2}~+~1\) by direct substitution.
Big Picture
What this problem is really about
To verify a proposed complex zero, substitute it for every occurrence of the variable and simplify the polynomial exactly. Reduce powers of the imaginary unit using i squared equals negative one; the proposal is verified only if the final value is zero. For a real-coefficient polynomial, the conjugate-zero theorem then supplies the matching nonreal partner.
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