A student thinks of a polynomial p(x) of arbitrary degree, and non-negative integer coefficients. How can you determine the student's polynomial by asking for two values of her polynomial, say p(a) and p(b), where a and b are positive integers? Hint: A positive integer n can be written uniquely in base k, where k is a positive integer. This follows directly by a repeated application of the division algorithm.
Let's let the unknown coefficients of the student's polynomial be c_i (meaning "c sub i"), so that
p(x) = sum( c_i * x^i ) for i = 0 to infinity
OK, we need to think of values for a and b which will allow us to reconstruct her polynomial. I conclude that the problem is assuming we can specify a, and then choose b given p(a), especially given the hint. In fact, we can't even choose a and b at the same time and be able to reconstruct p(x) from the response. For example, if we got the response p(a) = ab + a^2, and p(b) = ab + b^2 (whatever those numbers work out to be), we wouldn't know if the student had p(x) = ab + x^2, or p(x) = (a+b)x.
We want to find the values of c_i. So let's choose a, get an answer, and then choose b. The idea will be to choose an a such that we can then choose a b and be guaranteed of knowing the c_i. I'll go through the logic first, then the motivation in figuring ...
This solution shows, with justification, how to discover a polynomial given only two evaluations of the polynomial. It lays out why the solution works, and the thought process involved in arriving at the solution.
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