What's polynomial composition useful for?

It might be very useful if you need to make a substitution in an integral. Say you need to substitute $y=x^2+1$ to make part of the integral simpler, and then you need to do the same in another part of the integrand, which happens to be another polynomial.

Composition of polynomials is usually used when we have two polynomial functions, say $f(x)$ and $g(x)$, and we wish to perform something like:

$$f(g(x))=(f\circ g)(x),$$

If we needed to know the result of this for some value of $x$, we could of course simply compute $g(x)$ and then use this as our argument $x$ in $f(x)$.

However, say we are writing a computer program, and we need to calculate $f(g(x))$ several thousand times over the course of the program. Computing $g(x)$ for each value of $x$ first, and then computing $f(g(x))$ based on the output of $g(x)$ can be expensive. A much more efficient approach would be to sit down and actually perform the composition of the two polynomials (which will result in a polynomial of degree $\deg{f} + \deg{g}$), $(f\circ g)(x)$.

Of course, the difference in efficiency here would be fairly negligible, but it is just a simple example of where you could encounter polynomial composition.

EDIT: Incorrect. The output degree is $\deg{f} \times \deg{g}$. In general, the composition written out contains much more terms than the two separate, so it is not useful to expand them. In fact, there are algorithms that explicitly rely on iteration or composition because it is an efficient and numerically stable way to construct polynomials of very high degree.

Iteration is a special case of composition (namely, composition with itself). Studying $$f(x), f(f(x)), f(f(f(x)))...$$ is an entire field, see for example Mandelbrot sets and Julia sets. Now, a special case of the latter is Newtons method on solving polynomial equations numerically: every time you ask your calculator to numerically solve a polynomial equation, it will most likely use repeated polynomial composition somewhere in the algorithm. (This is a bit rough, what really is done is a bit more involved).