For a controller $C(s)$ and plant $G(s)$ connected by unity negative feedback,

$$U(s)=C(s)E(s),\qquad E(s)=R(s)-Y(s),$$

and

$$Y(s)=G(s)U(s).$$

Solving these relations gives the closed-loop transfer function from reference to output:

$$\frac{Y(s)}{R(s)}=\frac{C(s)G(s)}{1+C(s)G(s)}.$$

The product

$$L(s)=C(s)G(s)$$

is the loop transfer function.

The denominator

$$1+L(s)=0$$

is the characteristic equation. Its roots are the closed-loop poles and determine the natural modes of the feedback system.

This is the central algebraic effect of feedback: the controller does not merely scale the plant output. It changes the denominator and therefore changes the system dynamics and stability.