Imaginary or Surds Roots

If $\alpha+i\beta$ is a known imaginary root of a real-coefficient quartic, the Complex Conjugate Root Theorem (§3.4.1) hands us $\alpha-i\beta$ for free — so $\big(x-(\alpha+i\beta)\big)$ and $\big(x-(\alpha-i\beta)\big)$ are both factors, and hence so is their product:

Dividing the original polynomial by this real quadratic factor leaves a lower-degree quotient, solvable by known techniques. The identical idea works for a known surd root $p+\sqrt q$ of a rational-coefficient equation (§3.4.2): its automatic conjugate partner $p-\sqrt q$ gives the quadratic factor $\big(x-(p+\sqrt q)\big)\big(x-(p-\sqrt q)\big)=x^2-2px+(p^2-q)$. …