For every positive integer n, consider all monic polynomials f(x) with integer coefficients, such that for some real number a
x(f(x+a)−f(x))=nf(x)
Find the largest possible number of such polynomials f(x) for a fixed n<1000.
Details and assumptions
A polynomial is monic if its leading coefficient is 1. For example, the polynomial x3+3x−5 is monic but the polynomial −x4+2x3−6 is not.
500
Alestair no point posting wrong answers. Anyways stop posting brilliant problems. Anyways since the live period is over, here is a hint: show that for a fixed n the number of polynomials is the number of divisors of n.
To find the largest possible number of polynomials satisfying the given condition, we can analyze the equation step by step.
Given: x(f(x+a)−f(x)) = nf(x)
First, let's substitute x = 0 into the equation:
0(f(0+a)−f(0)) = nf(0)
0 = nf(0)
From this equation, we can conclude that either n = 0 or f(0) = 0.
Case 1: n = 0
In this case, the equation becomes:
x(f(x+a)−f(x)) = 0
For any value of x, this equation will hold true if either f(x+a) = f(x) or f(x) = 0. Thus, we can have infinite polynomials in this case.
Case 2: f(0) = 0
In this case, the equation becomes:
x(f(x+a)−f(x)) = nf(x)
xf(x+a)−xf(x) = nf(x)
xf(x+a) = (n+1)f(x)
Now, let's consider the degrees of the polynomials. Since f(x) is monic, its degree can be denoted as d. The degree of f(x+a) will be d as well. The degree of f(x+a) can be greater than d if and only if the leading coefficient of f(x+a) is a nonzero integer (since a polynomial with a leading coefficient of zero is not monic).
Thus, we have the following conditions:
1. If n = 0, any polynomial with degree d can be a solution.
2. If n ≠ 0, the degree of f(x+a) is at most d.
Now, let's determine the degree of f(x). We have:
xf(x+a) = (n+1)f(x)
Comparing the degrees of both sides, we get:
degree(xf(x+a)) = degree((n+1)f(x))
1 + d = d
This equation implies that d = -1, which means that f(x) is a constant polynomial (degree 0).
In summary:
- If n = 0, any polynomial of degree d can be a solution, where d is a positive integer. The number of possible polynomials in this case is infinite.
- If n ≠ 0, the polynomial f(x) is a constant polynomial. The number of possible polynomials in this case is 1.
Therefore, the largest possible number of polynomials satisfying the given condition for a fixed n < 1000 is either infinite (if n = 0) or 1 (if n ≠ 0).