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Conjugate Pair Theorem

Nonreal zeros of a real-coefficient polynomial always come in conjugate pairs $a \pm bi$ .

The Conjugate Pair Theorem: if $f$ has real coefficients and $a+bi$ ( $b \neq 0$ ) is a zero, then $a-bi$ is also a zero. Nonreal zeros of real polynomials never show up alone.

Examples

If $2+3i$ is a zero of a real-coefficient polynomial, $2-3i$ must also be a zero.

See also

Irrational Root Theorem conjugate

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