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