Finding polynomials with their values at points 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 $(p(x)=2x^3+x^2+3)$, 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.

A polynomial of degree $n$ has $n+1$ coefficients.  If you know the degree but don't know anything about those coefficients, you'll need  $n+1$ values of the polynomial to determine them.  If you don't know the degree, no amount of values of the polynomial will suffice.
