For every positive integer n consider all polynomials f(x) with integer coefficients, such that for some real number a *x(f(x+a)−f(x))=n*f(x) Find the largest possible number of such polynomials f(x) for a fixed n<1000?
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To find the largest possible number of such polynomials f(x) for a fixed value of n < 1000, we can analyze the equation:
a * x * (f(x + a) - f(x)) = n * f(x)
Let's break down the problem step by step:
1. Start by understanding the equation: We have a polynomial f(x) with integer coefficients, a real number a, and an integer n. The equation states that for any value of x, the expression on the left side of the equation is equal to n times f(x).
2. Simplify the equation: Expand the brackets on the left side and simplify to get a new equation. Rearrange the terms to isolate f(x) on one side:
a * x * f(x + a) - a * x * f(x) = n * f(x)
a * x * f(x) + a * x * f(a) - a * x * f(x) = n * f(x)
a * x * f(a) = n * f(x) (1)
3. Identify the constraints: We need to find the largest possible number of polynomials f(x) that satisfy equation (1) for a fixed value of n < 1000. Therefore, we need to consider the constraints on f(x) and a.
4. Analyze f(a): From equation (1), we can see that f(a) must be an integer, as a * x * f(a) = n * f(x). Therefore, f(a) should always be divisible by a for every value of x.
5. Analyze f(x): Since f(x) has integer coefficients, let's consider an arbitrary polynomial f(x) = c₀ + c₁ * x + c₂ * x² + ... + cₐ * x^a, where c₀, c₁, c₂, ..., cₐ are integer coefficients.
Substituting this polynomial into equation (1), we get:
a * x * (c₀ + c₁ * (x + a) + c₂ * (x + a)² + ... + cₐ * (x + a)^a) = n * (c₀ + c₁ * x + c₂ * x² + ... + cₐ * x^a)
6. Analyze the degree of the polynomial f(x): Notice that the left side of equation (1) is a polynomial of degree (a+1) in terms of x, while the right side is a polynomial of degree 'a'. For the equation to hold true for all values of x, both sides must have the same degree. Thus, (a+1) = a, which is not possible. Therefore, there are no such polynomials f(x) that satisfy the equation for every value of x.
7. Conclude: Based on the analysis, there are no polynomials f(x) with integer coefficients that satisfy the given equation for any fixed value of n < 1000.
Hence, the largest possible number of such polynomials is zero.