Know the Fundamental Theorem of Algebra and verify it for quadratic polynomials.
Find the other root of a quadratic with real coefficients when one complex root is given
Problem
A quadratic has real coefficients and \(\text{one}~\text{root}\) is \(3+4i\). What can you conclude about another root?
Big Picture
What this problem is really about
A real-coefficient polynomial pairs every nonreal root with its complex conjugate. We’ll confirm the imaginary part is nonzero, keep the real component fixed, reverse only the imaginary sign, and check that the pair’s sum and product are both real.
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