Also, does it relate to the degree of the polynomial in any way? I am struggling to get a high-level understanding of the characteristics of different degrees of polynomials - for example, their shape etc. In my textbook, as another example, it says that we can use cubic Hermite interpolation polynomials to construct a cubic spline. Why not a quadratic Hermite interpolation polynomial to build a quadratic spline? What would be ridiculous about the last statement?
Assuming your coefficients are real numbers (or rational, or integers, or complex), then the number of times you can differentiate a polynomial before reaching the $0$ polynomial is the degree of the polynomial plus $1$. Each time you differentiate, you get a result which has a degree $1$ lower than the one you started with, and only when you reach a degree $0$ polynomial (a constant) can you differentiate to get $0$.
And the problem with quadratic splines is that you don't have enough degrees of freedom. You can't freely set
- Value at the start point
- Value at the end point
- Derivative at the start point
- Derivative at the end point
if you only have a quadratic to work with. A quadratic has three coefficients, and thus only three pieces of information (like the value at some point, or the value of the derivative at some point) is enough to fully specify one. If you try to cram in four pieces of information (like the list above), you will most likely not be able to find a function which satisfies all of them.
A cubic, on the other hand, has $4$ coefficients, and thus has room for four separate pieces of information.