Refresher - The Conjugate Pairs Theorem

The Conjugate Pairs Theorem

If a polynomial

${\displaystyle p\left(x\right)=a_n{x}^n+a_{n–1}{x}^{n–1}+•••+a_2{\left(x\right)}^2+a_{1}x+a_0}$

has real coefficients, then any complex zeros occur in conjugate pairs. That is, if ${\displaystyle a+bi}$ is a zero then so is ${\displaystyle a–bi}$ and vice-versa.

Example:

${\displaystyle 2–3i}$ is a zero of ${\displaystyle p\left(x\right)=x^3–\left(3x\right)2+9x+13}$ as shown here:

By the conjugate pair theorem, ${\displaystyle 2+3i}$ is also a zero of ${\displaystyle p\left(x\right)}$.