Find the largest possible number of distinct integer values {x_1,x_2,…,x_n}, such that for a fixed reducible degree 4 polynomial with integer coefficients, |f(x_i)| is prime for all i?

To find the largest possible number of distinct integer values {x_1, x_2, ..., x_n}, such that |f(x_i)| is prime for all i, you need to consider the properties of reducible degree 4 polynomials and prime numbers.

First, let's understand what a reducible degree 4 polynomial is. A reducible polynomial of degree 4 can be factored into two or more polynomials of lower degree. It means the polynomial can be written as the product of two or more polynomials.

To find the largest possible number of distinct integer values that satisfy the given conditions, you will need to consider the factors of the polynomial and prime numbers.

Here's a step-by-step process to determine the largest possible number of distinct integer values:

1. Factorize the reducible degree 4 polynomial: Write the polynomial as a product of two or more lower-degree polynomials. For example, if the polynomial is f(x) = (x - a)(x - b)(x - c)(x - d), where a, b, c, and d are constants.

2. Identify the factors: Each factor should be set to zero to find the possible values of x. Solve each equation individually, such as (x - a) = 0, (x - b) = 0, (x - c) = 0, and (x - d) = 0 to obtain the respective values of x. These are potential values that satisfy the polynomial equation.

3. Test the values: Substitute each potential value of x into the polynomial f(x) and calculate the absolute value of f(x), |f(x)|. Check if the absolute value is a prime number. If it is, then that particular value satisfies the condition.

4. Count the number of distinct integer values: Identify the largest possible number of distinct integer values that satisfy the condition. Count the values obtained from step 3.

For example, let's suppose the reducible degree 4 polynomial is f(x) = (x - 2)(x + 1)(x - 4)(x + 5). In this case, we have four factors (x - 2), (x + 1), (x - 4), and (x + 5).

Setting each factor to zero, we find potential values of x:
(x - 2) = 0 => x = 2
(x + 1) = 0 => x = -1
(x - 4) = 0 => x = 4
(x + 5) = 0 => x = -5

Now, substitute these potential values of x into f(x):
For x = 2, f(2) = (2-2)(2+1)(2-4)(2+5) = 0
For x = -1, f(-1) = (-1-2)(-1+1)(-1-4)(-1+5) = 0
For x = 4, f(4) = (4-2)(4+1)(4-4)(4+5) = 0
For x = -5, f(-5) = (-5-2)(-5+1)(-5-4)(-5+5) = 0

In this case, all the calculated values of f(x) are not prime numbers. Therefore, there are no distinct integer values satisfying the condition for this example.

Repeat this process for other reducible degree 4 polynomials to determine the largest possible number of distinct integer values that satisfy the given conditions.