Complex Conjugates and Polynomial Roots

This lesson explores the complex conjugate and the powerful conjugate root theorem, which tells us how complex roots of polynomials with real coefficients must come in conjugate pairs. Understanding conjugates is essential for solving polynomial equations and for simplifying complex expressions.

The Complex Conjugate

If $z = a + bi$ , the complex conjugate (or simply conjugate) of $z$ is:

$\bar{z} = a - bi$

The conjugate is obtained by changing the sign of the imaginary part.

$z$	$\bar{z}$
$3 + 2i$	$3 - 2i$
$-1 - 4i$	$-1 + 4i$
$5$	$5$
$7i$	$-7i$

Properties of the Conjugate

For any complex numbers $z$ and $w$ :

Complex Conjugates and Polynomial Roots

Complex Conjugates and Polynomial Roots

The Complex Conjugate

Properties of the Conjugate

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