Is there any way I can find a polynomial given any 2 points (with x coordinate OF MY CHOICE): Let's say there's some polynomial I don't know(p(x)=2x3+x2+3), but my machine will give me an output. I give one x value of my choice, and it returns p(x), where p(x) is the polynomial function. I give another value of my choice., x+h, and get the output p(x+h). Given these outputs, I have to find p(x) as a polynomial.
What I've done is plugged in 0, which gives me the final term of the polynomial that is not multiplied by any power of x. Then I plug in 1, getting another output. When I find the "slope" of the two points, I get the sum of all the coefficients of all the terms that are powers of x. If I do this for the given p(x), I get 3, which is the sum of 2 and 1. However, I can't figure out what powers of x there are and what specific coefficients there are. Does anyone know how to solve this?
@GerryMyerson and @Shash said I can find the polynomial given the bound of the coefficients. I am confused as to what that means. There is only one number that is the sum of the coefficients. How is there a bound? Also, how do I find this sum of coefficients with just one value? I need to use one more value, M+1, as Shash said, so I can't use 2 values to find the max/sum, as I won't be able to ask for a value that is M+1. Can anyone help? Thanks.
EDIT: Non-negative integer coefficients are assumed.