Know the Fundamental Theorem of Algebra and connect it to polynomial roots.
Evaluate and classify a real or complex root candidate
Problem
Use direct substitution to determine whether \(i\) is a root of \(f(x)~=~x^{2}~+~1\).
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What this problem is really about
The root test is exact: substitute the candidate for every occurrence of the variable and simplify the resulting polynomial value. Keep the complex number parenthesized and reduce powers of the imaginary unit with i squared equal to negative one. A value of zero verifies a root; any nonzero value rules it out and is also the corresponding division remainder.
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