Suppose I have 3 cubic bezier curves a,b,c. Each of it was continuous from a to b and b to c at the endpoint of previous and starting point of next, both position and tangent. It is smooth curve
Then those 3 curves suddenly look like it is actually one polynomial of some degree
Is it always been a polynomial curve? And if it is then are there easy way to find out 1 polynomial function merged from those 3 cubic functions, or any number of cubic functions merged this way?
On
No.
The first point is governed by points $a,u,v,b$, the second by $b,w,x,c$. The tangents at $b$ coincide if $w-b=b-v$ (or possibly with a positive scalar factor sprinkled in). Apart from that, all points are "free". That simply gives us too many degrees of freedom to have a single combined cubic, and a higher degree polynomial will not work because the degree is detectable locally within both part curves (by seeing that all sufficiently high derivatives vanish)
On
No. Two different polynomials can only coincide on a finite number of points (and this number cannot exceed the highest degree).
Usually, Beziers that are continuous with continous tangent have a discontinuous curvature. This is avoided in the cubic spline curves. But the derivative of their curvature is discontinous.
Let's just deal with the case of two curves, $A$ and $B$.
It might be possible to represent the combined curve as a single cubic, or it might not. If $A$ and $B$ were designed independently, it's highly unlikely that they will be joinable in this way. But some design systems have a function for splitting a curve into several pieces, and if $A$ and $B$ were produced by this sort of splitting process, then it will be possible to join them back together into a single cubic.
If the joined curve can't be represented as a single cubic, then it can't be represented by a single polynomial curve of higher degree, either. In other words, increasing degree doesn't help.
Your question was "is joining always possible?". The answer is no.
A related question could be "is it sometimes possible?" The answer is yes.
I don't know if you're interested in the "sometimes" question. If you are, leave a comment, and I will supply more details.
There are several approaches to the "sometimes" question. One simple technique is to compute derivatives of orders one, two, and three of each curve at the common end-point. If these derivatives are all equal, then the two cubics are the same, and can be joined into one. Just having the same tangent or the same first derivative is not enough.