Where how and when to use polynomials?

I'm not sure I understand what it means to "solve polynomials" (to simplify them, or what?). However:
Imagine that you need a smooth curve to connect some points. Why? For the beauty to it is not broken ) That point in the format (x,y): (0,0),(1,1),(2,0),(3,1), here is the solution. This is one of an infinite number of solutions, among others it stands out because it is the minimum degree polynomial satisfying the condition. This is a very well - to multiply and stack the processor will be much faster than calculating, say, the sinuses, and therefore can very quickly calculate the height of the point on the curve for any X.

You would have asked why you need the numbers or the subtraction (because it is the same addition). For example, you need to add a lot of different variables multiplied together in varying degrees. Stupidly, you can manually write the appropriate code, and you can just use the formula of the polynomial.

For example, the polynomials (polynomials) are used for the polynomial hash, and they can be quickly multiplied by using fast Fourier transform.
So if your data is representable in the form of polynomials, you can quickly multiply, which is very useful.

If you wonder how fast computers consider functions (sine, logarithm...), it is possible that this approximation by Chebyshev polynomials.