fundamental theorem of algebra

Fundamental Theorem of Algebra

Let $a_k\in \mathbb{C}$ for $k=0,1,2,\dots n$, such that $a_n\ne 0$. Then the polynomial $P:\mathbb{C}\to \mathbb{C}$ defined by

$$\sum _{k=0}^n{a}_k{z}^k=a_n\prod _{k=1}^n(z-z_k)$$

has exactly $n$ roots $z_1,z_2,\dots ,z_n\in \mathbb{C}$, counted with multiplicity (i.e., some roots may be repeated).
In other words, every non-constant polynomial with complex coefficients has at least one complex root.
→ $\mathbb{C}$ is algebraically closed

What does the fundemental theorem of algebra state?

Every polynomial can be factored like

$$\sum _{k=0}^n{a}_k{z}^k=a_n\prod _{k=1}^n(z-z_k)$$

i.e. every polynomial has exactly $n$ roots $z_n$, counted with multiplicity.
→ Every non-constant polynomial with complex coefficients has at least one complex root → $\mathbb{C}$ is algebraically closed

What's special about writing polynomials in the form $c_n{\prod }_{k=1}^n(x-x_k)$? Why is $c_n$ in the front?

$c_n$ is necessary to match the leading coefficient. The product ${\prod }_{k=1}^n(x-x_k)$ is monic: multiply it out and the top term is $x^n$ with coefficient $1$. So on its own it can only produce polynomials whose leading coefficient is $1$. $c_n$ scales it to the actual leading coefficient $a_n$.
Keeping every factor monic ($x-x_k$, not $c_{k}x-d_k$) means the roots $x_k$ are read off directly, and all the scaling collects into the single constant $c_n$ out front rather than being smeared across the factors.