Complex Polynomials as a Product of Linear Factors

The Fundamental Theorem of Algebra says a degree-

$n$

polynomial has exactly

$n$

roots once you allow complex numbers (counting repeats) — so it always factors completely into linear pieces:

$p(z) = a(z - r_1)(z - r_2)\cdots(z - r_n)$

If the polynomial's coefficients are all real, any non-real root

$r$

comes paired with its conjugate

$\bar{r}$

as another root.

Example:

$z^2 - 2z + 5$

has roots

$1 \pm 2i$

(complete the square:

$(z-1)^2 = -4$

), so it factors as

$(z-(1+2i))(z-(1-2i))$

.