Further complex numbers · manipulative

Conjugate roots come in pairs

For a real-coefficient polynomial, non-real roots appear as conjugate pairs on the Argand plane. Drag z — its twin z̄ is forced so every coefficient stays real.

z (drag) z̄ twin real root r

Drag z off the real axis · the twin mirrors across the real axis

Quadratic factor q(x) = (x − z)(x − z̄)

Real root

−1
Break the twin
drop z̄ and keep only z — watch coefficients go complex

Working