Factor theorem

If $z=\alpha$ is a root of the polynomial $f(z)$ (i.e. $f(\alpha)=0$), then ${(z-\alpha)}$ is a factor of $f(z).$

Fundmental theorem of algebra

Working in complex numbers give us a fundamental algebraic result: that a polynomial with degree $n$ has $n$ roots.

The polynomial equation $a_nz^n + a_{n-1}z^{n-1} + \ldots + a_0 = 0$ has $n$ complex roots (including multiplicity).

Conjugate root theorem

The conjugate root theorem helps us solve polynomial equations more efficiently for cases with only real coefficients.

If $z=a+b\mathrm{i}$ is a root of the polynomial equation $P(z)=0$ with real coefficients, then the complex conjugate $z^* = a-b\mathrm{i}$ is also a root.

Examples