Solving for Prime Numbers in the Expression (n^2 - 2n - 8)

When dealing with polynomial expressions, determining the values of n for which the expression (n^2 - 2n - 8) becomes a prime number can be intriguing. In this article, we will explore the method to find such values and the underlying algebraic principles.

Reformulating the Expression

To start, let's rewrite the given polynomial expression:

n2 - 2n - 8 can be expressed as (n - 4)(n 2).

According to the properties of prime numbers, for (n - 4)(n 2) to be a prime number, one of the factors must be 1 and the other must be the prime number itself. This leads us to consider two cases:

Case 1: (n - 4 1)

Here, we can solve for n as follows:

n - 4 1

n 1 4

n 5

Substituting n 5 back into the original expression:

52 - 25 - 8 25 - 10 - 8 7

Since 7 is a prime number, n 5 is a valid solution for the expression to be prime.

Case 2: (n 2 1)

Here, we solve for n as follows:

n 2 1

n 1 - 2

n -1

Substituting n -1 back into the original expression:

(-1)2 - 2(-1) - 8 1 2 - 8 -5

Since -5 is not a prime number, n -1 is not a valid solution.

Additional Cases

It is also important to consider the case where the factors could be negative primes:

If n - 4 -1, then n -3. Substituting n -3 back into the original expression:

(-3)2 - 2(-3) - 8 9 6 - 8 7

If n 2 -1, then n -3. This case is already considered and yields a prime number. If n - 4 -1, then n -5. Substituting n -5 back into the original expression:

(-5)2 - 2(-5) - 8 25 10 - 8 27

If n 2 -1, then n -3, which we have already considered.

Conclusion

After thorough analysis, the only value of n that satisfies the condition for n^2 - 2n - 8 to be a prime number is:

n 3

Thus, the only valid solution is:

x 3

This exhaustive exploration ensures that all possible scenarios are considered, providing a comprehensive answer to the problem.