Theorem. If a complex number z0 is a root of a polynomial equation with real coefficients, then its conjugate z0 is also a root.
Proof. For P(x)=anxn+⋯+a0 with every ar real and P(z0)=0: conjugating both sides, and using ar=ar for real ar, gives P(z0)=anz0n+⋯+a0=P(z0). Since P(z0)=0=0, also P(z0)=0. ■
Every real number is technically also complex, so a genuinely non-real number α+iβ (β=0) is more precisely called a non-real complex number (sometimes just imaginary). The informal phrase "complex roots occur in pairs" really means: the non-real roots of a real-coefficient equation always pair up as conjugates.
Consequence for parity. Since non-real roots of a real-coefficient equation always come in pairs, an odd-degree real equation always has an odd number of real roots (at least one), and an even-degree real equation always has an even number of real roots (possibly zero).
Building a minimal equation from one known imaginary root. If α+iβ is a root of a real-coefficient equation, then (x−(α+iβ))(x−(α−iβ))=x2−2αx+(α2+β2) is automatically a factor — dividing this quadratic out of a higher-degree equation is the standard way to reduce it step by step.
Worked illustration. The monic minimum-degree real equation with root 2−3i: conjugate 2+3i is also a root; sum =4, product =22+32=13; equation x2−4x+13=0.