The so-called “Bezier Curve” is a method of representing a smooth curve in computer graphics.
Note that it is a method of curve representation, and not a curve in the usual sense in math contexts.

In computer graphics, there is a need to represent a smooth curve, in particular, finding a representation of a curve of a few given points entered/drawn by the user.
One way is to represent a curve by a set of points that lies on the curve. This method
obviously has some flaws. For example, it requires a lot collection of
points to represent a curve smoothly, and it will be again jagged
after magnification.

The Bezier Curve representation is a method to represent a curve
between 2 given points, by a polynomial parametric formula, with the
additional idea of using a few “control points” that specifies the
tangency of the given points.

The Bezier Curve method is named after the engineer
Pierre Bézier.

Given 4 points P0, P1, P2, P3, find a parametric formula in
polynomial such that it passes P0 with tangent vector[P0,P1], and
passes P2 with tangent vector[P3,P2].

Answer:
written in complex number notation:
z[t]=(1-t)^3*P0 + 3*t(1-t)^2*P1 + 3*t^2(1-t)*P2 + t^3*P3
We verify that
z[0]==P0, z'[0]==3*(P1-P0)
z[1]=P2, z'[1]==3*(P2-P3)